/
ISBN: 0-444-87823-8
Text
North-Holland Mathematical Library
Board of Advisory Editors :
M. Artin, H. Bass, J. Eells, W. Feit, P. J. Freyd, F. W. Gehring,
H. Halberstam, L. V. Hormander, J. H. B. Kemperman, H. A. Lauwerier,
W. A. J. Luxemburg, F. P. Peterson, I. M. Singer and A. C.
Zaanen
VOLUME 4
Cubic Forms
Algebra, Geometry, Arithmetic
YU.I.MANIN
Mathematical Institute V.A. Steklov
Academy of Sciences of the U.S.S. R.
Moscow
Translated from Russian
by
M. Hazewinkel
Second Edition
1986
© ELSEVIER SCIENCE PUBLISHERS B.V, 1986
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording or otherwise, without the prior permission of the copyright
owner.
First edition 1974
Second edition 1986
ISBN: 0444 878238
Published by:
ELSEVIER SCIENCE PUBLISHERS B.V.
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Library of Congress Cataloging-in-Publication Data
Manin, IU. I.
Cubic forms.
(North-Holland mathematical library ; v. 4)
Translation of: Kubicheskie formy.
Bibliography: p.
Includes index.
1. Surfaces, Cubic. I. Title. II. Series.
QA573.M2513 1986 516.3'6 83-20463
ISBN 0-444-87823-8 (U.S.)
Printed in the Netherlands
PREFACE TO THE SECOND EDITION
In the ten years since this book was published in English, there has been important
progress in a number of topics related to its subject. Were this book ю be written anew,
its title could be Algebraic Varieties close to the Rational Ones. Algebra, Geometry,
Arithmetic. In fact, this class of varieties has crystallized as a natural domain for the
methods developed and expounded in Cubic Forms.
In this edition the original text is left intact, except for a few corrections, but an
Appendix is added together with a list of references to original papers, mainly of the last
decade.
This Appendix sketches some of the most essential new results, constructions and
ideas, including the solutions of the Luroth and Zariski problems, the theory of the
descent and obstructions to the Hasse principle on rational varieties, and recent
applications of К-theory to arithmetic. Proofs are omitted since their complete
presentation would demand a new book. Meanwhile, this modest report will hopefully
be of use.
I am deeply indebted to V. E. Voskresenski and M. A. Tsfasman for their help in
preparing this second edition.
Moscow, 1984.
Yu. I. Manin
PREFACE TO THE SECOND EDITION ....................................v
CONTENTS........................................................vii
INSTRUCTIONS TO THE READER....................................... x
INTRODUCTION . . ............................................... 1
CHAPTER I. CH-QUASIGROUPS AND MOUFANG LOOPS 6
1. A survey of definitions and results......................... 6
2. Symmetric Abelian quasigroups................................11
3. CH-quasigroups...............................................15
4. Commutative Moufang loops ...................................21
5. The connection between CH-quasigroups and Moufang loops ... 25
6. Morphisms of CH-quasigroups and Moufang loops................28
7. The first structure theorem................................ 30
8. The second structure theorem................................ 33
9. Finite Fischer groups........................................34
10. Unsolved problems and bibliographical remarks................39
CHAPTER IL CLASSES OF POINTS ON CUBIC HYPERSURFACES 42
11. Admissible equivalence relations: a survey...................42
12. Unirationality...............................................46
13. Universal equivalence........................................54
14. R-equivalence: the basic properties..........................61
15. R-equivalence and quadratic extensions ......................65
16. Universal equivalence over local fields. Examples............69
17. Bibliographical remarks......................................76
CHAPTER III. TWO-DIMENSIONAL BIRATIONAL GEOMETRY 77
18. The main results.............................................77
19. Monoidal transformations.....................................82
20. Monoidal transformations and divisors...................... 90
21. The main theorems on birational maps.............. • . . . 100
22. Bibliographical remarks.....................................110
CHAPTER IV, THE TWENTY-SEVEN LINES 112
23. A survey of the results........................................112
24. Del Pezzo surfaces . . . . . """. : : . . . . . . . . 117
25. The Picard group and root systems..............................126
26. Exceptional curves and Weyl groups . . ........................134
27. The zeta function..............................................143
28. Minimality and classes of conjugate elements in Weyl groups . . .151
29. A cohomological invariant and the degree of unirationality. . . .154
30. Rational points................................................162
31. Tables and comments. Calculation of Я1. The theorem of Artin
and Tate......................................................... 174
32. Bibliographical remarks........................................182
CHAPTER V. MINIMAL CUBIC SURFACES 184
33. A survey of the results........................................184
34. The fundamental birational invariant..................... . .189
35. A bubble space................................................ 195
36. Calculations on cubic surfaces.................................200
37. Birational non-triviality......................................202
38. Birational classification......................................204
39. Relations between the generators...............................206
40. Bibliographical remarks........................................219
CHAPTER VI. THE BRAUER-GROTHENDIECK GROUP 220
41. A survey of the results. Obstructions to the Hasse principle . . . \ 220
42. The construction of Azumaya algebras...........................229
43., Brauer equivalence............................................234
44. The finiteness theorem.........................................237
45. Calculations for Brauer equivalence. Examples .................243
46. A negative result............................................ 264
47. Counter-examples to the Hasse principle........................276
48. Bibliographical remarks........................................283
APPENDIX. ALGEBRAIC VARIETIES CLOSE TO THE RATIONAL ONES.
ALGEBRA, GEOMETRY, ARITHMETIC
Introduction.................................................... 284
1. Galois cohomology, Picard groups and birational geometry .... 285
2. The Hasse principle and descent on rational varieties ..........288
3. Geometry of rational surfaces. Complements.................... • 294
4. The Luroth problem and the Zariski problem in dimension 3 ... 302
5. Rational points and equivalence relations...............306
6. Cubic surfaces and commutative Moufang loops (CML)......309
References (for the Appendix)............................313
REFERENCES.................................................318
AUTHOR INDEX...............................................323
LIST OF SYMBOLS............................................324
SUBJECT INDEX............................................. 325
1. The first sections of all chapters can be read consecutively, independent-
ly of the remaining text. These sections contain a survey of the main concepts
and results of the book, as well as some motivation and examples.
2. Interdependence table of the chapters:
(Dashed arrows indicate a weak dependence.)
3. Some standard notation:
Z the integers,
Q — the rational numbers,
R ' the real numbers,
C — the complex numbers,
Qp — the field of p-adic numbers,
Zn — the cyclic group of order n.
A The list of references, the author index, a list of the most frequently
occurring symbols, and the subject index can be found at the end of the book.
0.1. Every mathematician who is not indifferent to number theory has
felt the charm of Fermat’s theorem on the sum of two squares of natural
numbers. A psychologist of the Jungian school would probably think that
such diophantine problems are archetypal to a high degree.
The basic idea for the book presented here arose from an attempt to find
out what happens in the case of sums of three rational cubes. Needless to say,
the result is not nearly so simple, fundamental and complete as the classical
pattern. The author has generalized the problem along all the lines which oc-
curred to him, and has used all technical resources known to him. He obtained
as a result the multitude of non-associative composition laws, monoidal
transformations and Galois cohomologies which make up this book.
0.2. The problem of the sum of three cubes has a respectable history. The
basic result by the classical mathematicians is the following (see Dickson [1]):
Theorem. Every rational number is a sum of three rational cubes.
First proof (Ryley (1825); Richmond (1930)):
/ a3-36 f /-a3 + 35a+36\3 / З3в2 + З5в ?
a - I------------I + I--------------I + I--------------1 ,
'32<z2+34a+ 36 > \32а2+34а+3б/ \32д2 + 34я + 36/
This proof is simple, but not too illuminating. It would be nice to know what
lies behind this identity.
Second proof: After having added an extra coordinate Tq, we can write the
equation in homogeneous form:
aT3 + 7’3 + 7’3+7’| = 0.
This is the equation of a smooth cubic surface V in a three-dimensional projec-
tive space. There are ‘trivial’ rational points on this surface, e.g. (0, 0, 1, U).
Unfortunately these lie in the plane at infinity and do not give a solution to
our original problem. However, rational points on a cubic, surface can be multi-
plied, that is, we can construct new ones, starting from known points.
The first idea. Let x E V be some rational point. Construct the tangent plane
to V at x and let us denote by C(x) its intersection with V.
‘Generally speaking’, C(x) is an irreducible cubic curve in this plane with x
as a double point. Through x we draw all the lines in rational directions which
are tangent to К Each of these lines must intersect the cubic curve C(x) in three
points (counting multiplicities); but the intersection at x has multiplicity two,
which leaves only one point. The coordinates of this point are necessarily ra-
tional. In fact, the coordinates of the intersection points in terms of the pa-
rameters of the equations of the line are the roots of a cubic equation with
rational coefficients. This equation has a double rational root, corresponding
to x; therefore the third root is also rational. After this one can apply the
same procedure to the rational points of the curve C(x) and so on.
Unfortunately, for x = (0, 0, 1,-1) the curve C(x) consists of three lines
which are conjugate over Q, and there are no rational points on it except for x.
The second idea. In this case we draw a line in an arbitrary rational direction.
We only take care that the two other points of intersection of this line with F,
say у and у, do not coincide and that the curves C(y) and C(y) are ‘good’ as
described above.
Then the previous argument on cubic polynomials shows that y,y are defined
and conjugate over some quadratic extension К of the field of rational num-
bers Q.,(It can happen by accident thaty, у even have rational coordinates,
but then the problem is solved.)
As above, we construct ‘many’ points on C(y) with coordinates in K. Take
one of those points, say z, construct its conjugate z, and draw the line through
z and z. Because z and z are conjugate, we can assume that the coefficients
of the parameter equation of this line are rational. The third (besides z and z )
of its intersection points with 7, which we denote by z о z, then also has ra-
tional coordinates (by the same argument on cubic polynomials).
Of course, if we start with z =y, we simply return toy о у =x, but it is not
difficult to show that other points z E C(y) give many new rational points in-
cluding points which do not lie in the plane at infinity,
0.3. Although the second proof is considerably longer than the first, it con-
tains, in embryo form, interesting possibilities for establishing approaches to-
wards getting a review of all solutions of the equation, instead of giving only
an existence proof.
Multiplying points by means of C(x) gives an infinite family of solutions
of our diophantine equation. These solutions depend on as many independent
parameters as desired (the directions of the lines which occur in the construction).
Can all solutions be covered by a finite number of such families? How many
parameters are sufficient for this?
The composition law x о у is not defined everywhere (e.g., what isx о x?);
all the same it permits us to obtain new solutions from old ones. Is it possible,
by combining this composition with the construction of points on C(x), to
obtain all solutions out of a finite number of them?
0.4. A positive answer to the latter question is known; not for cubic surfaces,
but only for cubic curves (without singular points), зауяТ^+Т^ + =0.
This is the famous Mordell—Weil theorem on elliptic curves. The algebra, geom-
etry and arithmetic of cubic curves (one could add analysis, e.g. theta functions)
constitute a vast and actively developing field; see the survey of Cassels [3]
in the list of references.
The natural more-dimensional generalizations of elliptic curves, however,
are the Abelian varieties (and homogeneous spaces over them) and in general
not the cubic (hyper) surfaces. Nevertheless, it turns out that over non-closed
fields (in particular over number fields), there is a whole series of results from
the theory of elliptic curves which admit non-trivial analogues in the theory of
cubic surfaces. (Sometimes the statement of the theorem carries over almost
verbatim, although the mechanics of the proofs in dimension 2 have nothing
in common with the one-dimensional case; see Section 33.)
Three fundamental parallels follow.
0.5. (a) The composition law xy ~ и о (x о y) (u fixed) on an elliptic curve
turns its set of points into an Abelian group. (As above, the point x oy is de-
fined by the property that x, у, x о у are on one line.) On a cubic surface one
can divide the set of points into classes such that these classes can be composed
in a unique way by means of lines through representatives. After this the com-
position law XY ~ Uo (To Y) turns the set of classes E into an ‘almost’ Abelian
group of exponent six. ‘Almost’ because this composition can apparently be
non-associative. A slightly weaker associativity condition than the usual one,
which can be successfully proved, defines on E the structure of a ‘commutative
Moufang loop’. This structure is studied in Chapter I of this book, and the com-
position of classes of points in Chapter IL
(b)i The translations by means of a rational point generate almost all of
the group of birational maps of an elliptic curve into itself (more precisely,
they generate a subgroup of finite index). In the two-dimensional case, the
translation by x defines analogously a birational map tx. v^x о v of a surface
into itself. These maps (and similar ones connected with quadratic extensions
of the base field) also generate a subgroup of finite index in the group of all
birational maps of V into itself; in any case, if V is minimal. The proof (with
substantial specifications) is contained in Chapter V.
(c) An algorithm for settling the question of whether there are rational
points on a given plane elliptic curve has up till now not been found. The
first necessary condition is that there exist points ‘everywhere locally’. This
being fulfilled, there is the second necessary condition that the so-called
Cassels—Tate form becomes zero. In Chapter VI it will be shown that the
second condition admits a quite general formulation, which is in particular
applicable to cubic surfaces. There we also obtain (rather restricted) results on
the qiK Jiuns formulated in section 0.3. They give lower estimates for the nec-
essary number of parameters and for the number of generators of the set of
points.
These three subjects also represent, respectively, algebra, geometry and
arithmetic. Analysis and topology could essentially complete the picture.
For instance, starting with dimension 3, the ‘intermediate Jacobians’ of
A. Weil appear. In a less traditional direction we can expect that the group
generated by the maps tx has interesting ergodic properties. All this is not
touched upon in this book.
0.6. A part of the results expounded here has been taken from the journal
literature, old and new (including papers of the author). Another part is pub-
lishedJhere for the first time, for example the discussion on universal equiva-
lence in Chapter II and almost all calculations of Chapter VI.
Mainly classical material is contained in Chapter IV, where the geometry
of the famous configuration uf the 27 lines and its generalisations and applica-
tions are studied. Chapter III presents the necessary preparatory information
on birational maps.
The required algebraic—geometric background of the reader increases mono-
tonically with the numbers of the chapters. In Chapter I, ip general no algebraic
geometry is required. To understand Chapter II, it suffices if the reader is
familiar with the lectures of Safarevid [2] (by no means to its fullest extent).
Chapters 1II—V require the mastery of approximately half of Mumford’s book
[2], if the reader is willing to take a series of theorems on trust. Finally in
Chapter VI already the ghosts of ft ale cohomology faintly stir. To understand
it, it is also necessary to have some acquaintance with class field theory (struc-
ture of the Brauer group for local and global fields).
I.R. SafareviC taught me the algebraic-geometric approach to number
theory. Around ten years ago he drew my attention to cubic surfaces. He
conjectured, in particular, that some non-associative structure must play a
role in the description of the set of rational points. When these structures
started to appear, I was assisted in dealing with them by B.B. Venkov, A.I.
Kostrikin and V. A. Belousov. The talks with V. A. Iskovskih on the Brauer
group have been very useful to me. Some of the results of Chapter II are due
to A. Bel’skii; he has also been of considerable assistance in preparing the
manuscript for printing. The identification of the root systems Rr in Chap-
ter IV has been done by means of a method communicated to me by P. Deligne
in a private letter. To all these persons I am deeply indebted.
The papers of Grothendieck [2], Segre [3] andChatelet [1] have most of
all influenced the formation of the new ideas of this treatise.
Moscow, 1969 — 1970
Yu. I. Manin
CHAPTER I
GH-QUASIGROUPS AND MOUFANG LOOPS
First Scene:
An open place. Thunder and lightning.
Enter three witches.
Shakespeare. Macbeth, Act I
1. A survey of definitions and results
In this chapter we introduce and study some algebraic structures which
emerge in the theory of cubic hypersurfaces. The first section contains a sur-
vey of those results which have immediate applications in that theory. I
strongly recommend to restrict oneself at first to this survey and to go on
directly to the second chapter, returning to the first when necessary. Here
we give the exact definitions, state the theorems and give some motivation;
the proofs are contained in the next sections.
Definition 1Л. A set E with a binary composition law E X E E: (x, y)
ojus called a symmetric quasigroup if it satisfies one of the following
equivalent conditions:
(i) The three-place relation L (x. у, z) : x о у = z is invariant under all per-
mutations of X, y, Z.
(ii) The following identities hold for all x, у E E'.
X о у = у О х ,
(I.l)
ХО (хоу)=у .
(1-2)
The equivalence can be verified immediately.
The following geometric example may serve as background and motivation
for this definition: let E be the set of non-singular points of an irreducible
cubic curve F, embedded in a projective plane over a field k‘, and let the rela-
tion L(x, y, z) be ‘the cycle x +y + z is the intersection of V with some line'
(counting multiplicities). Here condition (i) is geometrically obvious but con-
dition (ii) is easier to work with algebraically.
In this example the quasigroup E satisfies the following additional prop-
erty: let и EE be some fixed element; we introduce on E the new composition
law xy = и о (x о j?). Then E becomes an Abelian group with и as unit element.
As the structure of Abelian groups is well known, one usually prefers to work
with this (new) composition law.
An axiomatization of this situation leads to the following:
Definition 1.2. A symmetric quasigroup £ is called Abelian if it satisfies
one of the following equivalent conditions:
(i) There exists an Abelian group structure on E with composition law
(x, y) , and there is an element c EE such that x о у = cx~^y~^ for all
x, у EE.
(ii) For any element и EE the composition law xy - и о (x oj>) turns E into
an Abelian group.
The equivalence of conditions (i) and (ii) will be verified in the next section.
Let us now consider an irreducible cubic hypersurface V of dimension > 2
embedded in some projective space; let E be the set of non-singular points. The
three-place relation L(x, y, z) on V is defined as before in the case of a cubic
curve. It is symmetric. However, in general, it does not come from a binary
composition law on V. This has to do with two geometric circumstances:
(a) When dim V = 1, the point x о x is defined as ‘the third intersection
point with V of the tangent line to V atx’. But, when dim V> 1, there are
many tangent lines at x: they fill up a whole tangent hyperplane.
(b) When dim V> 1, there can be lines completely lying in V. For two
points on such a line it is impossible to find a third such that the set of these
three forms the whole intersection cycle with some line.
In the next chapter we shall avoid these difficulties by considering instead
of E a quotient set of E such that the induced relation of ‘collinearity’ comes
from a symmetric quasigroup composition law. We cannot guarantee that this
quasigroup will be Abelian, as in the one-dimensional case. However, any three
points of V are contained in the intersection of V with a plane. This intersec-
tion is a cubic curve. Hence, as the lines through these points and the points
derived from them all stay in this plane, we obtain the result that any set of
three elements of the quasigroup generates an Abelian quasigroup.
This justifies the following definition:
Definition 1.3. A CH-quasigroup (CH stands for Cubic Hypersurface) is
a symmetric quasigroup in which any three elements generate an Abelian sub-
quasigroup.
We shall now state the main known results on the structure of CH-quasi-
groups.
Let E be a CH-quasigroup. By analogy with the Abelian case we intro-
duce on A a new composition law xy = и о (x oj), where и is some fixed ele-
ment. It is a remarkable fact that the structure thus obtained has been intro-
duced before in non-associative algebra theory, and has been thoroughly
studied by Bruck [2].
Definition 1.4. A set E with composition law (x, у) >-> xy is called a commu-
tative Moufang loop (henceforth abbreviated CML) if it satisfies the following
axioms:
(i) Commutativity: xy - yx for all x, у G E.
(ii) Unit element: их = x for allx C.E.
(iii) Inverses: there exists a map E E: x x-1 such that x-1(xy) = у
for all x, у G E.
(iv) Weak associativity:
for three factors: x(xy) = x2y; (1.3)
for four factors: (xy) (xz) = x2(yz), (1.4)
x(y(xz)) = (x2y)z . (1.5)
Theorem 1.5.//E is a CH-quasigroup, then the composition law
xy ~ и о (x о у) turns E into a CMLt
The axioms for a CML, as introduced in Definition 1.4, are not indepen-
dent. For example, it is possible to deduce (1.3), (L4) and (1.5) from either
(1.4) or (1.5) alone. We have included these identities because they immedi-
ately show how close CML s are to Abelian groups; in particular, any Abelian
group is a CML. The structure theory for CML s, due to Bruck, specifies and
pinpoints this closeness. Before we formulate its basic results, we introduce
some auxiliary definitions concerning CML s.
Morphisms of CML s are defined in the obvious way.
Let E ly Ey be a CML morphism. Its image f(E) is a Moufang subloop in
E^. The kernel of/is defined as the inverse image of the unit element inEp
A sequence of CML’s and CML-morphisms,
/ s
\^E, >E—>E. -H ,
where g is a surjective morphism, and/identifies E2 the kernel of g, is
called an extension of E± by E^, frequently we shall denote this extension by
the CMLE. The kernel ofg defines completely the equivalence relation in-
duced by the mapg, exactly as in the theory of groups. The equivalence
classes are the cosets of E with respect to/(E2). We write E\ = E/fifEf).
A subloop of the CMLE which is the kernel of some morphism is called
a normal subloop of E.
Definition 1.6. Let E be any CML. Then the set
Z(E) = {x EEI x(yz) “ (xy)z, Vy, z EE)
is called the associative centre of E.
Lemma 1.7. For any CML E, the associative centre Z(E) ofE is an associa-
tive subloop (and hence it is simply an Abelian group) in E; Z(E)is also a nor-
mal subloop in E.
We can now state the main structure theorems.
Theorem 1.8. Let E be any CML. Then the quotient loop of E by its centre,
E/Z(E\ is a CML of exponent 3; that is, in E/Z{E) the identity x3 = 1 is satis-
fied.
Theorem 1.9. Let E be a finite CML of exponent 3. Then its centre is non-
trivial.
From this it follows in particular that any finite CML of exponent 3 is ob-
tained by successive extensions by Abelian groups of exponent 3; and its order
is hence of the form 3K.
Theorem 1 JO. Any CML of exponent 3 with a finite number of generators
is finite.
Thus any CML with a finite number of generators can be obtained by suc-
cessive extensions by an Abelian group and a chain of finite Abelian groups of
exponent 3: the non-associativity is caused by the non-triviality of these exten-
sions and in’ the 3-component. A non-associative CML has order at least
81 (Hall [2]; in that paper a different notation is used).
1.11. Theorems 1.9 and 1.10 are rather deep. There are two methods of
proving them that I know of.
The original method of Bruck [2] is based on a very complicated and in-
geneous inductive process and uses some hundreds of non-associative identities.
Here a different approach is offered, which is based on the fact that the connec-
tion between CH-quasigroups and Moufang loops can be reversed. The results
on the classification of CML s of exponent 3 will then follow from the classi-
fication of distributive symmetric quasigroups; that is, quasigroups which
satisfy
x о (у о z) ~ (x oy) o(xoz) .
This was observed by Belousov [1].
Theorem 1.9 can then be obtained as a corollary of a purely group theoret-
ical theorem due to Fischer [1]. The proof of this theorem, although it uses
considerably less calculations than the proof of Bruck, is far from simple and
uses such deep results of the theory of finite groups as the Brauer—Suzuki and
Thompson—Feit theorems! Moreover, this proof neither yields Theorem 1.10,
which therefore is not proved here, nor does it give an explicit estimate of the
length of a composition series with Abelian quotients for a CML in terms of
the number of its generators. Bruck’s method does furnish such an estimate,
though it is unknown how accurate it is.
I would like to bring this beautiful and subtle structure theory to the atten-
tion of the algebraists. There is a series of perplexing unsolved problems and
the proofs clearly can do with some improvements.
Whether there are really non-associative CML's in the algebraic—geometric
theory of the next chapters I have been unable to clear up. This seems very
likely, but it is not impossible that only Abelian groups occur. In that case
the main algebraic constructions of this chapter of course turn out to be super-
fluous for the sequel.
2. Symmetric Abelian quasigroups
In this section we make use of Definitions 1.1 and 1.2 (i). Our main result
will consist of the statement and proof of some criteria for the commutativity
of a symmetric quasigroup E. First we introduce some notation which will be
constantly employed in the sequel.
Let £ be a symmetric quasigroup. For an arbitrary element x G E, the
symbol tx : E -> E will denote the ^reflection" tx{y) = x о у. It follows
from formulae (1.1) and (1.2) that t^ = 1 and that = ty(x). We denote
by T(E) the group of permutations of the set E generated by all the tXi x EzE,
and by T°(£) we denote its subgroup consisting of the products of an even
number of reflections tx.
Theorem 2.1. Let Ebe a symmetric quasigroup. The following properties
are equivalent:
(i) E is an Abelian symmetric quasigroup.
(ii) The group T®(E) is Abelian.
(iii) Vx, y, z GE, (tx ty tz)2 = 1.
(iv) For any element и EE, the composition law xy ~uo (x о у) turns E
into an Abelian group.
(v) The same as (iv) for some fixed element и EE.
Under these conditions E is a principal homogeneous space over T®(E).
Proof. We shall verify the following implications:
(i)<===(v)
(i) => (ii). Let x о у = cx'^y^ in terms of the composition law of an Abelian
group structure on E Then
Mv(z) = = (x~~1y)z .
«л у
Thus pairwise products of reflections are group translations and therefore they
commute. From this it follows that T®(E) is commutative.
(ii) => (iii). Because the group T®(E) is Abelian, pairwise products of reflec-
tions commute, so that
(t t t -1 t (t t ) t t = (t t ) f t t t =1.
vx у zf xy^zx'yz z XJ x у у z
(iii) => (i). We first of all remark that for all x, y, z EE we have
*x fy lz=tW’ <L6>
where w = у о (x оzf In fact, for an arbitrary и EE,
*y lz = {х гу fu (z) = fu *y fx = u ° l У ° ° = w ° u = fw
(the second equality follows from (iii): (tx tylu)2 = 1 => txtytu = tutytx).
The commutativity of F now results from the following lemma.
Lemma 2.1.1. Let и EE be a fixed element of the symmetric quasigroup
E with property (iii) of Theorem 2.1. Then the map E -> T®(E) :x *+x ~tutx
is a one-to-one correspondence, T®(E) is an Abelian group, and for all x, у EE,
Xoy - U OU x~^ у . (1.7)
Proof. It is clear that the map x p* x is an embedding. By applying for-
mula (1.6) repeatedly to an arbitrary element of T®(E), it may be represented
as a product of just two reflections. Further, again by formula (L6),
where v = x о (и о у). Therefore x н» x is an epimorphism.
The commutativity of T°(£) follows from Theorem 2.1 (iii):
Гх tuty^tytutx^tutxtuty^tutytutx<>^yzzy^-
Finally, formula (1.7) is derived from formula (1.6) in the following way:
иоих У ~^иои^х^и^у^~ ^и^иои^х^уо(иои)^ ~ Wxoy ’
Here formula (1.6) is applied twice to the product of three reflections in paren-
theses. This concludes the proof of the lemma.
(iii) => (iv). We introduce in E the composition law xy = и о (x oy) and we
check the Abelian group axioms. The commutativity and the existence of an
inverse and of a unit element follow from the symmetry of the quasigroup E:
Uy = Uo(uoy)=y ,
xy = и О (x о у) - Ц о (у ох) ~ух ,
X"1 “ (и о и) о X ,
XX-1 = и О (х о (х о (и о и)У) = и .
The associativity follows from the commutativity of T®(E)\
(xy)z = uo(zo (xy)) = t t t t (y) ,
14 wV
x(yz) = uo(xo(yz)) = t t t t (y).
(iv) => (v). Obvious.
(v) => (i). We suppose that и о (x oj>) = xy is the composition law of an
Abelian group. The commutativity of E will follow when we have proved
the formula
X оу = (и ou)x~*y~l
And in fact this is equivalent to the identity
(ху)(хоу) = иои,
but
(xj>) (x О у) = U о [(м о (х о/)) о (х о у)] == М о М .
Finally, the last statement of the theorem follows from the observation
made in the course of the proof of (i) => (ii).
This proves the theorem.
As an application of the theorem, we give a simple geometric proof of
the commutativity of the quasigroup of points of a plane cubic curve. With
the usual definition, operating directly with the composition law uo(x oy)9
the associativity is extremely difficult to see.
Somewhat unexpectedly, it turns out that the equality (txty tz)2 = 1 has
a straightforward geometric interpretation.
Example 2.2. Let к be a field, V an absolutely irreducible cubic curve in
the projective plane over k. We denote by E the set of its non-singular Ap-
points. It is made into a symmetric quasigroup as described in Section 1. Take
three points x, y, z and an arbitrary point и on E (see Fig. LI). The equal-
ity (tx t tУ и - и means that by ‘reflecting’ и successively through x, y, z
and then again through x, y, z, we will arrive again at u. We shall show this.
We number the lines by means of which the successive reflections are car-
ried out starting with the line (uz). We denote by F) the union of the odd
numbered lines (they are drawn boldly in Fig. LI). This is a reducible cubic
curve. It intersects V in nine points, which are marked in the figure.
Fig. 1.1.
The curve K2 is the union of the three straight lines with even numbers;
obviously it passes through eight of these points, but possibly not through u.
To prove that U2 also passes through u, we note that the projective space
of cubic forms of three variables is nine-dimensional. Consequently, the sub-
space of those that become zero in eight points of the plane is, ‘generally
speaking’, one-dimensional. Therefore the equation of the curve F2 is a linear
combination of the equations of the curves У and Kj, that is to say, У2
passes through all the points of the intersection of V and Kp
This ‘quasi-classicaF argument has some flaws. One of these is hidden
in the expression ‘generally speaking’; this is partially.dealt with by a theorem
in Walker’s book [1 ]. The others are connected with the consideration of the
various cases of ‘not lying in general position’ of л:, у, z, и. Therefore it is prob-
ably better, ultimately, to consider Fig. 1.1 as a geometric illustration of the
theorem rather than to base the proof on it, although it shows the nature of
the relation (7X ty tz)2 = 1 very clearly.
We conclude this section with a corollary.
Corollary 2.3. Every subquasigroup and quotient quasigroup of an Abelian
quasigroup is again Abelian.
Proof. The symmetry is obvious and the commutativity follows from prop-
erty (iii) of Theorem 2.1.
3. CH-quasigroups
We denote by CK the class of symmetric quasigroups in which any к ele-
ments generate an Abelian subquasigroup. Evidently CK is contained in
For к > 4, CK coincides with the class of Abelian quasigroups. Indeed,
(rx ty tz)2 и ~ и for all x, y, z, и G E, where E G CK. because for к > 4, x, y, z, и
belong to an Abelian subquasigroup of E. Therefore (txtytz)2 - 1 within it
and property 2.1 (iii) gives the desired result.
The class is called the class of CH-quasigroups (cf. Definition 1.3). The
remaining part of this chapter is dedicated to the structure theory of C2. It
certainly contains non-Abelian quasigroups: the smallest order of such a quasi-
group is 81 (see Example 3.4). Possibly the structure of +he quasigroups be-
longing to C\ and C2 also deserves a study.
Our first result on the structure of CH-quasigroups is analogous to part of
Theorem 2.1.
Theorem 3.1. Let E be a symmetric quasigroup. Then the following prop-
erties are equivalent'.
(i) E is a CH-quasigroup.
(ii) Vx, у EE, tx ty tx Oy — ty Oy.
(iii) Let и EE be a fixed element. Then the map E T®(E): x^x-tutx
is an embedding and it has the following properties '.
x'oy -u~ou x у -1 [x,y] , (1.8)
where \x,у} - xу x у . Moreover,
и ou belongs to the centre ofT®(E) , (1.9)
[x, у ] commutes with x,y , (1.10)
[x,j]3 = l. (1.11)
Compare condition (iii) with Lemma 2.1.1. The group T°(E) can be non-
Abelian, but the properties (1.9), (LIO), (1.11) show that it is not too far from
being Abelian.
Proof, (i) => (ii). The equality txtytXOy(z) ~ tyoy(z) can be checked within
the subquasigroup generated by x, y, z. But this one is Abelian because E is a
CH-quasigroup: formula (1.6) then gives the desired result.
(ii) (iii). Here, and also further on, we shall use the notation j8(x) ~x о x.
We start by proving some identities.
Lemma 3.1.1. In any symmetric quasigroup E where the identity
Gr ty ^хоу ~ tp(y) holds, the following equalities are valid for all x, y, z EE:
(fyQ “ tpix) ’ (L12)
(x <7)0 z — x о (у о (j3(x) oz)) , (1.13)
fxfy ~ ГУ°РМ ~ O-14)
Proof. Because of the symmetry of the composition law, we have
^xoy txty — tTherefore tXQy — t^x^ty tx ~ t$(y)tx ty (ihe latter equality
holds because xoy ~y ox). Formula (1.12) follows from this.
Furthermore,
lx fx оу (2) “ о у - lx ly fax) ’
which is equivalent to formula (1.13).
Finally,
ty (z) = ( у о Z) о X = у о (z о (0(у) о х)) = ty tx oe(y)(z)
The second equality of formula (1.14) is obtained by taking the inverses of
both parts of the first equality and interchangings andy. This concludes the
proof of the lemma.
It is convenient for the statements of the following lemmas to call elements
a, b e T^(E) related if there exist elements x, y, z G E such that a = tx ty and
b-txtz. Evidently this is a symmetric relation.
Lemma 3.1.2. If E satisfies condition (ii) of Theorem 3.1 and if a, b G T\E}
are related, then я”1, b are also related.
Proof. Suppose a - tx ty, b = tx tz; because of formula (1.14),
a ~ tytx~~ txtyo@(x) ’
which proves the lemma.
Formulae (1.10) and (1.11) now follow from the next lemma:
Lemma 3.1.3. Under the same conditions as before, let a, b G T®(E} be re-
lated elements. Then [a, ~ 1 and [a, b} commutes with a and b.
Proof. According to Lemma 3.1.2, a and h-1 are also related. Therefore we
can assume that a~ tYtY, b = t~tY. Using formula (1.12), we find
д2й2 = ^у)Ч(х)Ч(х)Ч(г) = = (Йа)2 ’
Z>404 =t J J 2 C J 2 = (Z>a)4 .
e (X) e2(x) /32(y)
After multiplying the first equality on the left by zz and on the right by b, and
the second equality on the left by Z?-1 and on the right by zz-1, we obtain:
a3b3 = b3a3 = (ab)3 = (ba)3 .
From this it follows that
(ba)a3b3 ~ ba (ba)3 = (ba)2(ba)2 = a3b2a2b2 = a2(ab)2b2 = a3bab3 ,
or, after multiplying this chain of equalities by b~3a^ on the right,
ba3 - a3b; i.e., [b,a3] = 1. Because of symmetry, [zz, b3] = 1.
Because the pair zz-1, b is also related, we have a~3b2 = ba~3ba~3. Multi-
plying this equality on the left by zz3 and on the right by zz, and using [zz3, b] = 1,
we obtain zzZ?2zz = ba3b, that is, [zzZz, ba\ = 1.
Set c - [zz, 6] = zz^zz"1^"1. Then [c, ab] ~ [c. ba] ~ 1; moreover, с3 ~ 1, be-
cause (ab)3 ~ (cba)3 - c3(ba)3 ~ (ha)3.
Finally, we show that [c, = [c, zz] = 1. Indeed, a3b3 = (ba)3 - (c~^ab)3 =
= c(ab)3 = abcab. Multiply this equality by zz”1 on the left and by b~x on the
right to obtain ab - bca;but ab ~ cba; therefore [c, Z>] - 1, and by symmetry
[c, zz] = 1. This proves the lemma.
Corollary 3.1.4. The map (c, d) [c,d] on the subgroup ofT^(E) gener-
ated by a pair of related elements is a skewsyrnmetric bilinear form, that is,
[c, d] = [d, c]-1 and [c^, d] = [q, d] [c2, d].
This is a general group theoretic fact which follows from the centrality of
the commutator subgroup.
3.1.5. To prove formula (1.9), it suffices to check that tu t^ commutes
with all elements of the form tx tu> because these generate the group T®(E).
We set a = b=txtu. Then b2 = t0(u)t&(x), from which ab2 = tu
and further,
а(аЬ2ГХ = tu(txtu)2tu =eutx)2
~ f0(u) “ f0(u) - •
Consequently, [zz, b3] ~ 1, and as the commutator is bilinear and [zz, 63] - 1,
it follows from this that [a,b] = 1.
3.1.6. Finally, we derive the identity (L8). Combining everything which has
been proved, we have:
I op = tu txoy - tu ty tx
“ fx
= 0(u)x 2y -1 x -1 = 0(M)x-1 у-1 [x, y] .
(iii) ==> (i). Let E be a symmetric quasigroup which satisfies condition (iii)
of the theorem..Let u, x, у EE be any three elements. We denote by Eq the
subquasigroup generated by x, y, u, and we shall show that it is Abelian. The
image of Eq under the map zhz = tu tz clearly is contained in the subgroup
G C T°(E) generated by the elements 0(u), x, у; this follows from the fact
that according to formula (L8) the composition law о can be expressed in
terms of groups.
The commutator subgroup of the group G is in its centre and has exponent 3.
We introduce on G the new composition law
a + b ~ ab [a, b} .
We show that G with this composition law is an Abelian group G+. Indeed
a + b - ab [a, Z>] - ba [b, a] = b + a .
Further,
(a + b)+ c - ab [a, b] c[ab [a, b], c] - abc [a, b] \at c] [b, c] ,
a + (Z> + c) - abc \b, c] \a,bc[b, c] ] -abc [а, Л] [«, c] [b, c] .
(We constantly use Corollary 3.1.4.) The unit element and inverses in G
and G. coincide.
From formula (1.8) it now follows that for all xl} €E0,
x^ox^ ~ ио1л ~ *
Thus Eq can be embedded in an Abelian quasigroup constructed from the
group G+ and that means that Eq is Abelian. This concludes the proof of the
theorem.
Using this result we shall now describe one method of constructing CH-
quasigroups from groups which are given by generators and relations of a
special form. The structure of these groups has been studied by Fischer [1];
his results will be set forth in detail in Section 9. Here we only show that one
can construct non-Abelian CH-quasigroups by means of the Fischer groups.
Definition 3.2. A Fischer group is a pair (G, E) consisting of a group G and
a system of generators E C G satisfying the following conditions:
(i) Vx G E, x2 = 1.
(ii) Ух, у CE, (xy)3 = 1 andxyx GE,
From this definition one easily obtains some elementary properties of
Fischer groups:
3.2.1. The set E constitutes a full class of conjugate elements of G,
In fact, let x, у GE; the relation (xy)3 = 1 implies that у = (xy)x(xy)”1.
We set F ~ x U xE; this is clearly a system of generators in G, and for all
g GF we havegxg”1 GE, Therefore all conjugates of x are contained in E,
3.2.2. The commutator subgroup G/ of the group G is generated by prod-,
ucts of pairs of elements of E; moreover G/G' Z2 or {1}.
In fact, Vx, у GE, xy GG1 because x and у are conjugate. On the other
hand, x together with the products of pairs of elements of E generates G, and
x2 = 1.
3.2.3. The pair (GIN, EN/N) is a Fischer group for every normal subgroup
NCG,
Theorem 3.3. Let (G, E) be any Fischer group. We introduce a composition
law on E by setting
X о у = хух .
Then E becomes a CH-quasigroup, and the map E -> T(E): x ь» tx can be ex-
tended to an epimorphism of groups with as kernel the centre of the group G.
Proof. The symmetry of E follows from the fact that
X о у = хух = уху ~ у о х ,
X о (х оу) = х(хух)х =у.
The CH property is verified by means of property (iii) of Theorem 3.1:
tyOy(Z)=ty(Z')=yZy ’
гх(у^оу(2)=х(у(УхУ(2)УхУ)У)х=У2У
The last assertion of the theorem follows from the fact that tx acts on 2? as
conjugation by means of x.
This proves the theorem.
Remark. The identity relation j3(x) = x holds in E under the conditions of
Theorem 3.3.
3.4. Examples of non-Abelian CH-quasigroups. We fix an integer r > 4, and
construct a finite non-Abelian CH-quasigroup with r generators. Let B(3,r-1) de-
note the Burnside group of exponent 3, generated by ap..., ary with the iden-
tity relation x3 = 1. Take the semi-direct product G of the groups Z2 = {l,x0}
and В = B(3,r- 1) with В as normal subgroup on which x0 acts according to
xoatxo = C-
We set Eq = (xq, xoa^ . . Xgzzr_|). The product of a pair of elements of
Eq belongs to В and its third power is therefore equal to 1. Consequently Eq be-
longs to one class of conjugate involutions £; one easily convinces oneself that
(G, E) is a Fischer group and that Gf = B. It follows from Theorem 3.3 that E
with the composition lawx о у -xyx is a CH-quasigroup, while T®‘(E)~ B/Z(Gf
where Z (G) is the centre of G. But В is nilpotent of class 3 for r > 4 (a theo-
rem of Levi and Van der Waerden, cf. Hall [1]). The group T®(E) is therefore
non-Abelian, and consequently the CH-quasigroup E is non-Abelian (Theorem
2.1).
4. Commutative Moufang loops
We use the definition of a CML as stated in Definition 1.4. Before we start
with the proof of the main result of this section, Moufang’s theorem, we state
some elementary properties of a CML E.
The unit element и G E and the map x x-1 are uniquely defined: if
u^x - x = U2X> ^en because of property (iii) of Definition 1.4,
= (w1x)x“1 - (u2x)x"i -u^ ;
if x-ix = и -yx, then x1 -y. From this it follows immediately that
(x"1)"1 ~x and (xy)-1 ==x-1y_1. In factjX^fxy) ~y ^x"1 ~ (xy)-1y =>
=>X”ly 1 = (xy)”1 . .... .........~“™:.™............
Finally, we deduce the identities (1.3) and (1.5) from (1.4). Substituting
2 = u in eq. (1.4), we obtain eq. (1.3). Replacing^ byy-*1 in eq. (1.4), we obtain
x2(y-1z) = (y-1*) (xz). Multiplying both parts by у 2 and bringing a factory
in each of the four factors on the left and on the right according to formula
(1.4), we find (x2y)z = x(y(xz)), which is eq. (1.5).
Theorem 4.1 (Moufang’s theorem). Let x, y, z.be three elements in a CML
E for which the relation (xy)z -x(yz) holds. Then the subloop generated by
them is associative and hence is an Abelian group.
Corollary 4.1 J. Every two elements in a CML E generate an Abelian sub-
group; in particular, words in which only two variables occur may be handled
as if they belonged to an Abelian group, and one may disregard the distribution
of the parentheses.
In fact, {xy)u - x(yw), where и is the unit element, and x,y are arbitrary
elements.
The proof of Theorem 4.1 rests on two lemmas:
Lemma 4.1.2. Let E be a CML, x,y EE two elements. Then the map
aXy \E-*E defined by the formula
ax v<Z> =
-Sz
is an automorphism of E.
Proof. It is clear that .. is 1-1 and onto and that it takes the unit element
-Sz
into itself. We must only verify that for all v, wEEt
\y^w)=ax>y(v)aXty(w).
It follows from formulae (Г.4) and (1.3) that у(y(uw)) = (yv) (yw). Multi-
plying both parts by x2 and bringing a factor x in each of the factors on the left
and on the right according to formula (L4), we find
(xy) (x (y (vw))) = (x (yv)) (x(yw)),
from which
X (y(yw)) = (xy)-1 ( [x (J'U)] [x (J'W)] ) .
Multiplying both parts by (xy)-1 and on the right bringing a factor (xy)-1 in
each of the factors in square brackets according to formula (1.4), we find, fi-
nally,
aX,y(™) = aX,y(v)ax,y(w)-
This proves the lemma.
We define the associator of three elements in E by
(X,y,z)= [(xy)z] [x(yz)]-1 .
Lemma 4.1.3. Let the elementsx,yf z EE satisfy the relation (x, y, z) = 1.
Then this relations remains valid
(i) under replacing any of the elements x, y, z by its inverse;
(ii) under all permutations of x, у, z.
We say that elements x,y, z with this property behave associatively. Let
x, y, z, v be elements such that every three of them behave associatively. Them
(iii) (x, y, zv) = 1.
Proof, (i). By definition,
(x,y,z)=i (xy)z=x(yz)<*a(z) = z.
Because ax y is an automorphism, we have ax y(z~~1)~ z"1, and that means
(x, y, z"1) = 1. By symmetry, x can be replaced by x”1. The middle term needs
a bit more work,
(x,y,z) = 1 «y=z~1(x~1(z(xy))) = z~1{x~2[x(z(xy))]}
= z~1{x-2[(x2z)y]} *a _t(y)=y.
H "V* V* * v z
gt Л
Therefore the same argument as before shows that (x, j-1, z) ~ 1. This proves
the first part of the lemma.
(ii) . We note that x and z obviously can be permuted;The invariance with
respect to a cyclic permutation is verified as follows:
x-1 (yz) = X-2 (x(>z)] = x~2 [(xj-)z] = y(x~lz~) =>O, z,x”1) = 1
=* (y, z,x)=l .
Finally,
ax v(z) = z and ax v(y) = U =* ax v(zy) = ZV (X’ У’ zy) = 1 •
This proves the lemma.
4.1.4. Proof of Theorem 4.1. Consider a maximal subset F CE, containing
x, y, z, such that every three elements of F behave associatively. Such a subset
exists by Zorn’s lemma. According to Lemma 4.1.3, F is closed under multi-
plication and taking inverses and it is therefore an Abelian group. The ele-
ments x, y, z generate a subgroup of it. This proves the theorem.
4.2. From Lemma 4.1.3 and Moufang’s Theorem 4.1, the first part of
Lemma 1.7 follows: the associative centre of a CML is an Abelian subgroup.
The second part results from the following characterization of normal subloops.
Definition 4.2.1. A normal subloop H С E in the CML category is a subloop
which satisfies one of the following equivalent conditions:
(i) H is the kernel of some morphism f: E -+ F of CML s.
(ii) x (yH) ~ (xy) H for all x, у E E.
Proof of the equivalence (outline), (i) => (ii). Let H be the kernel of /; then
r(j-1(^-1((^)H)))=/(j)-1(/(x)-1(r(x)/(^)))= 1 ,
from which у~1(х~1((ху)Я))) С. H. The opposite inclusion is proved similarly;
one can also use the fact that ax y is an automorphism of E,
(ii) => (i). A subset vi E of the form xH, where x E E, is called a coset with
respect to H. It is easy to see that the cosets form a partition of E into disjoint
subsets. On the set of cosets E/H one can define a CML composition law which is
induced by the multiplication in E, so that the canonical map/? -+E/H is a
morphism with kernel H. The details are standard and are left to the reader as
an exercise.
4.2. 2. It is clear from Lemma 4.2.1 that the normal subloops of a CMLZT
can be characterized as those subloops which are invariant with respect to the
automorphisms ax y for allx,T E£. By analogy with group theory, these
automorphisms are called interior automorphisms. They generate a group of
automorphisms of E\ the elements of this group are also called interior auto-
morphisms.
5. The connection between CH-quasigroups and Moufang loops
In this section we shall prove the following two theorems:
Theorem 5.1. Let E be a CH-quasigroup and и EE an arbitrary element.
Then the composition lawxy = u о (x о у) turns E into a CML with unit
element u. One obtains isomorphic CMLs for different choices of u.
Theorem 5.2. Let E be a CML and с E Z(E ) an arbitrary element of its
centre. Then the composition law x о у = cx~ly"~l turns E into a CH-quasi-
group.
We note that here, in contrast with 5.1, one can obtain non-isomorphic
CH-quasigroups for different choices of c. The precise description of those c
for which they turn out to be isomorphic will be given in Section 6.
5.1.1. Proof of Theorem 5.1. The formulae xy -yx and их ~x are obvious.
The inverse element is found from the equation xx1 = u:
и о (x ox”1) = U ^x1 = j3(w) ox .
The identity x-1(xt) ~y is verified as follows:
(*У) = w ° ((j3(w) о x) о (и о (x оу)))
fU *(3(и)ах (U ° (* °X>) =Л tu (и о (X oy)) = y
in virtue of formula (1.14) .
Finally, we shall check formula (L4). We note that in any CH-quasigroup
the identity ^(z)fZOx ^(t) ~ *xOO holds, which can be rewritten as
(zox)o(zoy) = 0(z)o(xoy). (1.15)
Taking into account formula (1.12) we find
(Xy) (xz) = U О «И о (х о У)) о (U о (х о z))) = tu t0W(у о z)
= U о (Р(и) О (/3(х) о (у о z))) = U о ((и о /j(x))° (и o(yoz)))=X?(yz).
It remains to be shown that one obtains isomorphic CML s for different
choices of и € E, We set
Xy=Uo(xoy), X*y-Vo(xoy)}
and we define a map f: E -+E by /(x) = и о (и о х). It is clear that / is bijective
and that/(w) = v. It is therefore sufficient to verify that/(xy) =/(x) *f(y).
In fact,
f(xy) = tu tvtu (x oy) ,
f(x) *f(y) =>tv(f(x) o/(y)) = Z/pfr) W* о J)
=^Ул>2(*о^)=
This proves the theorem.
We shall now list three useful supplements to it.
5.1.2. Under the conditions of Theorem 5.1, the operation x oy can be
represented in terms ofxy and the element (3(u) asxcy = ^(w)x"1y”i.
In fact,
0(н)(ху)-1 = и о (P(u) о (0(u) о (xy))) = U о (ху) =Х о у .
5.1.3. The element /3(u) is in the centre of the CML with composition law
uo(xoy).
In fact,
0(m) (xy) = м о (/3(M) о (и о (x о у))) = tu tfi(u)tu ty (x) ,
(0 (u)x)y = Uo((uo(P(u)ox))oy) = tuty tu tpw (x) .
It now remains to apply the result 3.1.5, according to which is in the
centre of the group T®(E).
(These observations show that the constructions of Theorems 5.1 and 5.2
are inverse to one another.)
5.1.4. The identity relation p(x) ~ x in a quasigroup E is equivalent to the
relations x3 = w and p(u) = и in the corresponding CML.
Indeed, for all x,
[3(x) = x 0(м)х”2”Х =>x3 = Д(м)=>х3 - /З(и) = м3 = u.
Conversely, if x3 = и for all x and 0(u) = uwe have, in virtue of 5.1.2,
0(x) = 0(m)x~2 = x~2 -x.
5.2.1. Proof of Theorem 5.2. It is clear that x о у = у о х. Further,
х о (х оу) = сх^гх^у”1)”1 = у,
as in an Abelian group, because с, x, у behave associatively.
It remains to check that the identity of Theorem 3.1 (ii) holds. We have
again as in an Abelian group. But with the left part one must be a little careful
because it depends on four elements:
x oy = cx^y”1 ,
<z) = 1хУ^2 1)=(^)z * >
ty 1хоу (Z) = = C(y~2z)x-1-,
according to eq. (1.5). Finally,
{x ‘y fxoy (z) = cx~1(e~lx(y2z-1')) = A4.
This proves the theorem.
6. Morphisms of CH-quasigroups and Moufang loops
We describe the functorial properties of the constructions of the preceding
section. First of all it is clear that any morphism f: E -> F of CH-quasigroups
induces a morphism of the corresponding CMLs with unit elements и EE,
v E F if and only if /(w) = /(u).
The inverse question is more difficult and more interesting. Let7i, F be two
CMLs, and let c EZ(E), d E Z(F). Construct the two CH-quasigroups Ec and
Fd with the composition laws cx~^y~^ and du~\~\ respectively. We shall
designate the unit elements of E and F by the symbol 1.
Theorem 6.1.Letf: Ec ~+Fd be a morphism of CH-quasigroups. Then
d =/(<?)/( I)2 and the map g = f(l)^ f: E F is a CML morphism. Here
g(c) = df(\y~\
Conversely, let there be given a CML morphism g : E ~>F and an element
bEF such that g(c) = db\ Then the map f= b~\g : Ec~>Ed is a morphism
of CH-quasigroups.
Before proving Theorem 6.1, we state two important corollaries:
Corollary 6.1.1. Let E be some CML, and let c, d EZ(Ef The quasigroups
Ec, Ed are isomorphic if and only if there exist an automorphism g : E -> E
and an element b EE such thatg(c) -db\
Example 6.1.2. Let E be an Abelian group of exponent 3. Its automorphism
group is transitive on the set of non-zero elements. Therefore from E one can
obtain exactly two non-isomorphic CH-quasigroups Ec: one for c = 1 and the
other for с Ф 1.
Corollary 6.1.3. The category of CH-quasigroups with the identity relation
0(x) = x and a distinguished element is equivalent to the category of CML s
of exponent 3.
The functor which establishes this equivalence transforms a quasigroup E
with distinguished element и into the CML with the same underlying set E
and the composition law xy = и о (x oy). This CML has exponent 3 according
to 5.1.2 and 5.1.4, and the composition law xoy can be reconstructed asx-1^”1.
Theorem 6.1 then shows that the morphisms of quasigroups which preserve
the distinguished elements are exactly the morphisms of the corresponding
CMLs.
6.1.4. Proof of Theorem 6.1. Let /: Ec -*Ed be a quasigroup morphism.
Then
ficx-1 y~l) = , (Ы 6)
from which we obtain f(c) = d/(l)”2 by taking x =y = 1. Substituting x ~ 1
and x = ^respectively, in eq. (1.16), we find
Г(сх-1) = с?/(1)-1Г(хГ1,
from which
fix)-1 =d-1/(l)/(Cx-1), (1.17)
ЯХГ1 =fWT2f(y-y (1-18)
Substituting eqs. (1.17) and (1.18) in the right-hand side of eq. (1.16), we find
ftcx-'y-l) = (ЛОЯсх-1)) (ГОГ2 Ку~1)У) .
Replacing ex”1 by x and y~l by у, we obtain
Using the associativity formula (L5) in which x ^/(l)”1 H-/(y) and
/(x), we obtain finally
from which, using eq. (1.4),
g(xy) f(xy) =/(l)-2 (f(x)f(yS) = g(x)g(y) .
This proves the first part of the theorem.
Conversely, let a CML morphism g : E -+ F be given and let g(c) = db\
We set /= b~^g, it is clear that/(l) = h”1. We must verify the identity
f(cx~xy~^ = df(x)-x f(y)~x .
We have
f(cx~xy~l) = b~xg(cx~1y~l) = i-1((</h3)(g(x)-1 g(j)^1)) •
Since d is in the centre of F, we obtain from this
f(cx~vy~l) = d(b2(g(x)-lg(y)-1)).
On the other hand,
df(x)~l fty)-1 = </(ftg(x)-1)(bg(y)_1) =d(b2(g(x)-1g(j’)_1),
again because of formula (1.4). This concludes the proof.
7. The first structure theorem
Definition 7.1. Let E be a symmetric quasigroup. Then E is called a distribu-
tive quasigroup if it satisfies one of the following equivalent conditions:
(i) T(E) is a group of automorphisms of К
(ii) x о (у о z) = (x о 7) о (x о z) for all x, y, z E E (distributivity).
(iii) £ is a CH-quasigroup with the identity relation 0(x) ~ x.
Proof of the equivalence, (i) (ii). Conditions (i) and (ii) are equivalent be-
cause the group T(E) is generated by the reflections tx.
(ii) => (iii). Substituting z = x in relation (ii) and multiplying both parts by
x oy, we obtain Д(х) = x. Consequently,
P(x)o (yoz) = (xoy)o (xoz) = A (z) = t t(z)=> t tt ~t^x.
\ J j \ } *0(X) УУ 7 XOy Xy J ХОу^Х у p(x)
Theorem 3.1 (ii) then shows that E is a CH-quasigroup.
(ii i) ==> (ii). This follows from formula (1.15).
7.2. Now let E be any CH-quasigroup, и EE a fixed element. As in Section
3, we consider the embeddingE-> T°(£): x ь*х = tu tx. In addition, let
g : T®(E) G be some group homomorphism. We denote by E* the image of
E in G under the composed map x »->g(x).
There is a unique quasigroup structure on£' such that E ~+Er is a morphism
(of quasigroups). In fact, the composition law on E' C G can be expressed in
group theoretical terms by Theorem 3.1 (iii), and all relations then transform
into group theoretical relations. The homomorphism g obviously preserves
these relations.
We apply these considerations to the natural homomorphism
g : T°(£) -> T\E)jZ, where Z is the centre of the group T°(£).
Theorem 73. As in 7.2, let us denote by El the image of the composed map
E -» T°(£) 5- T®(E)IZ. Then the following assertions hold'.
(i) The equivalence relation on E induced by this map and the quotient
quasigroup E’t the image of E, do not depend on the choice of u.
(ii) The quasigroup E’ is distributive.
(iii) All fibres of the morphism E -+Er are Abelian quasigroups.
Proof, (i).
g(fu Q = ^u Vx = {u fy <mod Z) ** fx ly e Z ’
and this condition clearly does not depend on the choice of u.
(ii) . It is clear that E’ is a CH-quasigroup; in virtue of Definition 7.1, it now
suffices to verify that g(0(x)) == g(x) for all x € E. And in fact, according to
Theorem 3.1 (iii),
(x) = fi(u) x “2 = x ~2 (mod Z)^x (mod Z) ,
because j3 (w) G Z according to formula (1.9); and by formula (LI 1) and
Corollary 3.1.4,x 3 CZ.
(iii) . If#(x)=g(j9, then
g(x^y) = /3(g(x))^g(Mx)) = 4x) .
Therefore the fibres of the mapx ^g(x) are closed with respect to composition.
They are Abelian because if x, у belong to one and the same fibre F С E,
then txty belongs to the centre of the group T®(E), so that the group T®(F)
is Abelian.
This proves the theorem.
We shall now show that the interpretation of Theorem 7.3 in terms of
CMLs leads immediately to a proof of Theorem 1.8.
Theorem 7.4. Let E be a CH-quasigroup, и GE a fixed element, and let
f: E -» El be the quasigroup morphism constructed in Theorem 7.3. We in-
troduce on E a CML structure with multiplication xy=uo(xoy) and on Er
a CML structure with multiplicationx'y' ~f(u)ofx'oy). Then f is a CML
morphism with identifies E’ with the quotient loop of E by its centre. More-
over, has exponent 3.
Proof. It is clear from the definition that for all x, у GE, we have x {y)~xy.
Therefore the group T®(E) coincides with the permutation group of E which
is generated by the multiplication by all possible elements of E in the sense
of the CML composition law. In particular, the image of x coincides with the
image of и in Ef if and only if x belongs to the centre of T°(£), that is,
for ally, z GE,
x(y(z))=y(x(zj)ox(yz) = (xz)y.
By definition, this means that the kernel of f coincides with the associative
centre of E.
The last assertion of the theorem follows from the distributivity and
Remark 5.1.4.
8. The second structure theorem
In this section we shall show that the study of distributive quasigroups,
and therefore of CML s of exponent 3, to a large extent results in a purely
group theoretical problem. More precisely, the construction of CH-quasigroups
by means of Fischer groups (Definition 3.2) permits us to obtain all possible
distributive quasigroups.
Theorem 8.1. Let E be a distributive quasigroup. We embed E in T(E) by
identifying the element x EE with tx G T(E). Then the pair (T(E), E) is a
Fischer group, and the composition law on Eis given, as in Theorem 3.3, by
the formula
t ~ t t t = t t t
xoy x у x уху *
Proof. Formula (L14) gives (taking into account the identity 0(x) =x):
txoy = txtytx = tytxty^^txty>>3=1 •
This proves what we wanted.
Remark 8.2.1. Using 3.2.2, we obtain from this that for a distributive quasi-
group E the group T°(£) is the commutative subgroup of T(Ef The centre of
T(E) is trivial because of Theorem 3.3.
Remark 8.2.2. Let I be some set. We construct the free distributive quasi-
group generated by I. To this end we set G(7) - F(I)/N, where F(I) is the
free group generated by /, and N is the normal subgroup generated by the
words x2 and f°r alix,.y G/ and s2eF(7). Let
E(I) C G(T) be the full class of conjugate elements which contains the image of
I. It is clear that (G(/), E(Fj) is a Fischer group, so that there is a natural distrib-
utive quasigroup structure on E(I). E(T) is free in the following sense: for
every distributive quasigroup £ and set of elements {xj, i EI, there exists a
morphism £(/) ”>£ which sends the image of i EI into x, EE. The proof is
left to the reader as an easy exercise.
I do not know whether a free Fischer group G(I) with a finite number of
generators is finite. From Theorem 3.3, Bruck’s Theorem 1.10 and Remark
5.1.4, it follows that in any case the quotient group of G(/) by its centre is
finite. A direct proof of this fact would in turn subsume Theorem 1.10. This
is an interesting group theoretical problem of the Burnside type.
9. Finite Fischer groups
The basic aims of this section are the proofs of Theorems 9.1 and 9.2,
which are due to Fischer. By combining them with earlier established results
we immediately obtain Theorem 1.9 on the structure of CML s of exponent 3.
Theorem 9.1, Let (G, E) be a finite Fischer group. Then the following three
assertions are equivalent:
(i) The commutator subgroup G1 is nilpotent.
(ii) G' is a 3-group, and G/G' = Z2-
(iii) E co nsists of 3K elemen ts.
9.1.1. Proof, (ii) => (i). Obvious.
(ii) =► (iii). The group G acts transitively on E by conjugation; the centralizer
2G(x) ={у €G\xy=yx}
is the isotropy subgroup of an arbitrary element x EE and it contains the ele-
ment x of order two. The order of G is equal to 2 • 3m\ consequently the order
of ZG{x) is equal to 2 • 3^, from which IEI = (G: ZG(xf) = (From now
on we shall denote by \M I the order of the set M)
(i) =* (ii). Because the group G’ is nilpotent, its only Sylow 3-group G^
is a direct summand. Because (G: Gf) = 2 or 1, G$ is the only Sylow 3-group
in G and is therefore a normal subgroup. But for all x, у E E we have
(xj>)3 = 1 ^xy EG^. Therefore the group G/G$ is generated by one involution,
which is obviously non-trivial because 2 divides I Gl and 2 does not divide IG$ I.
(iii) =► (ii). This part of the proof lies deeper. We need two lemmas:
Lemma 9.1.2. Let N GG be a subgroup, N and let N C\E be the
subgroup generated by N П E. Then we have for every element x GN О E,
N = {NQE}Zn(x).
Proof. N acts on E by conjugation. The orbit of x GN HE under this is equal
to TV О E. Indeed, it is contained in TV П E; on the other hand,
‘ у GNCiE =>y = (x oy)ox = (x O y)x(x oy) ,
andxoy eN HE together withx, y. Therefore (TV: ZN(xj) = iNCiEl. Since
the pair ({TVnE},TVClE)is also a Fischer group, applying the same considera-
tions to it, we obtain
(^HE}:Z{Arn^(x))= l{TVnE}HEl= ITVHEI.
On the other hand, by an elementary theorem.(Hall [1], 1.5.5) we find that
the indices of the pairs of groups marked by (1) and (2) in the diagram
N
{NOE}Zn(x)
^(x)
(2) /
are related by the inequality (1) > (2). It follows immediately from this that
(TV :{TV Л E}Zy(x)) = 1, which proves the lemma.
Lemma 9.1.3> Let x GE and IE I > 1. There exists a Sy low 3-subgroup
HGG such thatxHx -H.
Proof. Let у GE, у ^x, and let {x,y} C G be the subgroup generated by
x andy. Obviously {x,y}' = Zj. We denote by H the maximal 3-subgroup in
G with the properties
{x,y}'cH, xHx = H.
We show that Я is a Sylow subgroup in G. Otherwise it is contained in a larger
3-group, where its normalizer will be larger than itself. It therefore suffices to
check that a Sylow 3-subgroup F containing H of the normalizer
TV = {n G G\nHn = H} coincides with H. For this in turn it suffices to check
that F = HZF(x). In fact, xHZF(x)x = HZF(x)\ therefore it follows from the
maximality of H that HZF(x) = H.
The inclusion HZF(x) C F is obvious.
To prove the opposite inclusion, we first calculate {E C\N}1. We have
у EE C\N<>у EEt xyEH. (1.19)
Indeed,
у EE П N ^yHy -H => xyHyx = H{xy} is a 3-group;
moreover, xH{xy}x =H{xy}. From the maximality of H it then follows
that xy E H. Conversely, if xy E H and у E E, then yxHxy ~ H =* у EN.
The group {E Л N}' is generated by pairwise products of elements of
E Ci TV; it then follows from (1.19) that it is generated by some pairwise
products of elements of H. In particular, {E Ci N}’ С H, so that {E Ci N}'
is a Sylow 3-group in {E Ci TV}. But {E nN} is a normal subgroup in TV and ac-
cording to Lemma 9.1.2, TV = {E nN}Zj^(x}. Therefore there exists in TV a
Sylow 3-subgroup F generated by{E CiTV}' and the centralizer ZF(x). For this
we have F ~{E QN}f ZF(x) C HZF(x). This concludes the proof of the lemma.
9.1.4. Proof of the implication (iii) => (ii). We return to Theorem 9.1.
Let x EE, and let H C G be the Sylow 3-subgroup whose existence was proved
in Lemma 9.1.3. Considering the action by conjugation of G on E, we find as usual
IE I = (G : Zq{x)). Since IE I = 3K by hypothesis, we have
0? • (*)) “ (H{x}: Zp ^x\ (x)).
On the other hand,
IEI = (Я{х] : 2и{х}(хУ) = IE О H{x] I.
Therefore E C H{x} =>G = H{x}, which proves what we want.
Theorem 9.2. Let (G, E)be a finite Fischer group. Then the equivalent as-
sertions (i)—(iii) of Theorem 9.1 hold.
9.2.1. Proof. We proceed by induction on the order of G. We shall suppose
that the theorem is proved for Fischer groups of order < | G|. We distinguish two
cases, depending on whether there exists in G a minimal normal subgroup N
different from Gf.
Case 1. There exists a minimal normal subgroup NCG different from G
Step 1. The group G is solvable. Indeed, chooser GF ThenTV{x}=^ G.
Otherwise, G=NUNx^ECNx (because E И TV = 0)=> Gf GN. Therefore
{/V{x}TYF} is a proper subgroup in G, so that{TV{x] HF}' is nilpotent by the
induction hypothesis, and so{TV{x} AF}is solvable. It is clear that {TVjx} A F}n TV
is a normal subgroup in TV..
If there exists an x G F for which the intersection {;V{x} И F}n N is non-
trivial, then N has a non-trivial radical (that is a maximal solvable normal sub-
group). Because the radical of N is a characteristic subgroup, it is a normal
subgroup in G. From the minimality of TV it follows that the radical coincides
with TV, that is, that TV is solvable. By the induction hypothesis, G/TV is solvable,
and hence G is solvable.
If Vx GEf {TV{x} П E} A TV = {1}, then Vx GE, NxEE = (x). (In fact,
if j GTVx П F, thenyx G{TV{x} П F} HTV.) This means that there is a natural
isomorphism F ^xNE/N, from which TV C Z(G), because the action by conju-
gation of TV on F is then trivial. It follows from this that G is solvable because
G/TV is solvable by the induction hypothesis.
Step 2. The group G' is nilpotent. Since G is solvable, there exists a normal
subgroup TV C G such that where q is a power of some prime number
p. We note that if p Ф 3, then TV C Z(G). In fact, if TV £ Z(G), then there
are x, у GF such that x Vy and x GTVy; but then xy GN, so that I TV I = 0
(mod 3). We distinguish three possibilities.
(a)p#=2 or 3. Then TV C Z(G) andTVCG', because (l/Vl, I G/G' I) = 1. The
group (G/N)/(G'/N) ~ G/G' is Abelian, so that G’/N D (G/Nf. The group
(G/N)' is nilpotent by induction hypothesis. If G = Gthen G/N= (G/N)' =
= G'/N so that G' is nilpotent. If G Ф G\ then (G/N)/(G'/N) Z2 and again
(G/N)r = G '/N, so that G' is nilpotent.
(b) p = 2. We have again TV C Z(G). If N C G\ the same argument as in (a)
gives what we want. The case TV <£ G' is impossible since then G ~ TV X G'
G' = G" and (G/N)' = Gl is a 3-group by induction hypothesis; but then
F-(TV)^(1).
(c) p = 3. (G/Nf is a 3-group by induction hypothesis, so that G' is a 3-
group.
Case 2. G' is a minimal normal subgroup ofG. Following Fischer , we de-
duce from this a contradiction, provided that G' Ф Z3.
Lemma 9.2.2. Let Ebe a distributive quasigroup, and let x^xz be an auto-
morphism of order two (an involution) of E. We set Ez = {x EE\xz ~ x} and
for an arbitrary element x EEZ we set Ez x = {y EEly oyz = x}. The fol-
lowing assertions hold:
(i) Ez, EZXCE are non-empty subquasigroups.
(ii)Ei Xi Л Ezx* for ххФ x2, andE=\J
x<=Ez &z, x •
(iii) The Ez x are isomorphic for different x. In particular, \E I = \EZ I \EZX I.
Proof. (i).j> о yz G Ez for all ju G£; x EEZ x because of the distributivity.
Ez and Ez x are therefore non-empty. If x, у G Ez, then x о у G Ez because z is
an automorphism. Let у у y2 EEZ x; then y^ = x о y\ and y2 - x о. у 2, from
which у । о у 2 =x о о y%) and (j^ о j?2)° (jj ° y^f “ x, so that
У\°У2&Е2,х-
(ii) . Obvious.
(iii) . Letxj, x2 EEZ andx3 =Xj ox2 EEZ. Then the mapj>>-»x3 oy=x3j>x3
establishes an isomorphism between £_ and E7 Y .In fact,
J>1 °y\ =*1 **x3 о (jj ojj)=X2 «(x3 °7i)o(x3 ojp2 =x2 ,
Х3°(^1 °j2) = (x3°y])o(x3oj2) .
9.2 .3 Conclusion of the proof of Theorem 9.2. Let (G, E) be a finite Fischer
group in which Gr is a minimal normal subgroup and let z G G be any element
of order 2. It acts on E by conjugation.
We first show that there exists a z with \EZ I > 1. In fact, suppose that this
is not the case. If I Ez I = 1, Ez = (y), then x о xz = у, that is, xz = yxy for
allx G£, so that zy €Z(G) because G ~{E}. This does not contradict the
minimality of G' only if z = у G E. Thus E is the only class of conjugates
under the involution. For G =# Gwe have E Cl Gf ~ 0, so that G' has odd
order and thus is solvable by the Feit—Thomson theorem. This is a contradic-
tion provided that G' Z3. For G ~ G’, the group G must be simple. But any
two different elements x, у E£ belong to different Sylow 2-subgroups of G
because (xy)3 = 1. Therefore a Sylow 2-subgroup contains a unique element
of order two and that means that it is either cyclic or it is the generalized,
quaternion group (Hall [1]). This contradicts the fact that the group is simple
(theorem of Bumside, Hall [1], and the theorem of Brauer and Suzuki [1]).
Thus, let z E G be an involution with \EZ I > 1. Then also Ez ±E, because
otherwise z GZ(G), and the only possible case Z(G) = G' cannot occur in a
Fischer group. Thus Ezx E for all x E Ez (Lemma 9.2.2 (ii)). The groups
{Ez} and {Ez л} are Fischer groups of order < | G |; moreover, it is clear that
{Ez} OE=Ez and {Ez ПE = Ez x. From the induction hypothesis, Lemma
9.2.2 and Theorem 9.1, it then follows that I Ez I ~ 3a, \EZ x I = 3^, so that
1£ I = 3a*b in virtue of Lemma 9.2.2 (iii). Theorem 9.1 then shows that G'
is a 3-group, which again is only possible for Gf Z3.
This concludes the proof of Theorem 9.2.
9.3 . Proof of Theorem 1.9. Let £ be a finite CML of exponent 3. The compo-
sition law x о у = x"1y~1 defines a distributive quasigroup structure on it,
for which the group T°(£) coincides with the ‘group of multiplications’
(translations) of the CML E. According to Fischer’s theorem, T°(E) = T(E\
is a 3-group and so it has a non-trivial centre. Theorem 7.4 then shows that
the centre of the CML E is non-trivial.
10. Unsolved problems and bibliographical remarks
Symmetric quasigroups were introduced by Bruck [1] and CH-quasi-
groups were introduced by the author (Manin [6]); the investigation of their
structure theory was started in the same papers. Later, Venkov and Belousov
(see his paper [1]) indicated to the author the connection between CH-quasi-
groups and CMLs. The general case was analyzed in Manin [9]. Hall [2] has
obtained results similar to those expounded in Section 8; however, he used a
different notation. The proofs in Section 9 are taken from Fischer [1]. In the
latter paper, a more general case is dealt with; namely, the case where the
relation between the involutions has the form (xy)P = 1, with p prime.
An essentially different method (and the first) to study the structure of
CML s can be found in Chapter VII of Bruck [2]. In that book, the following
interesting unsolved problem is raised;
Let £ be a CML.
Definition 10.1. (i) Let E^~E\ is generated by associators of the
form (хг-, у, z), where хг- G E^ y,z<EE. The series E§ Э E^ D E2 2). . . is
called the lower central series of the CML E.
(ii ) E has nilpotency class i if E^ - (1).
Problem 10.2. What is the nilpotency class k(n) of the free CML with n
generators. What is its 3-order 7 (л)?
Obviously, Z(l) = 3, /(2) = 9, and Hall [2] proved that /(3) = 81.
Theorem 10.2.1 (Bruck [2], Chapter VII). 1 + [^] < k(n) < и-l for all
и > 3.
Obviously, A;(1) = k(2) = 1 (Moufang’s Theorem 4.1). The first unknown
case is: k(5) = 3 or 4? This is a very curious question in itself, but also in view
of the following theorem of Bruck [3]:
If k(5) = 3, then k(ri) = 1 + [|n] for all n>3.
The upper estimate for k(n) is obtained by Bruck by a complicated induction
on n, and the lower one by means of the following interesting example of a
CML.
Example 10.3. Let £ be a linear space over the field of three elements,
and let Л*£= ® A* L (its exterior algebra). Further we set E = L X A*£
and define on E the composition law
{a, x) (b, ^)' = (a + b, x + у + (x -y) ab) .
This turns E into a CML, (Why? Does there exist a more transparent explana-
tion than a formal verification of the axioms?)
The lower central series looks like Ei = (0, ф ^)> z 1 • The proof
is by induction on i, making use of the identity
((zz, x), (b, y), (c, z)) = (0, xbc + yea + zab).
In particular, if dim L ~ n > 3, and (xz-) is a basis of L, then the subloop E
generated by the generators (xz, 1), i = has nilpotency class [\n] + 1.
10.4. Let T^(E) be the group.of multiplications of a free CML£ with n
generators (7’°(£>) is generated by the maps E -> E: у >-+xy). Then its nilpo-
tency class is equal to ~ ~ '
t(n) - 2k(n)~ 1 .
Using this, one can translate Problem 10.2 into the language of Fischer groups.
This problem for n = 5 should lend itself to analysis on an electronic com-
puter.
CHAPTER II
CLASSES OF POINTS ON CUBIC HYPERSURFACES
11. Admissible equivalence relations: a survey
Throughout this chapter, we fix some infinite field k. Unless it is stated
otherwise, all algebraic varieties and schemes will be understood to be defined
over k\ and points will be geometric points with values in a sub field of some
fixed algebraic closure of k.
The main objects of study in this chapter are cubic hypersurfaces defined
over k. By definition, such a hypersurface V is given by a form of the third de-
gree: F(Tq, ..., Tn) = 0, where (To,..., Tn) is a homogeneous coordinate
system in some projective space over k. Unless it is stated otherwise, the form
F will be assumed to be irreducible over the algebraic closure of k\ that is
to say, the scheme Kis geometrically irreducible and reduced. (It is obvious,
however, that we shall also have to consider reducible hypersurfaces, e.g., in-
tersections of V with a tangent plane.)
Let be the set of non-singular ^-points of the cubic hypersurface
VC P\ (We recall that a point x ~ (rQ,..., tn) is non-singular on V if in the
affine coordinate system with origin atx the equation of Khas a non-vanishing
linear part; by putting this linear part equal to zero, one obtains the equation of
the tangent hyperplane to К at the point x.)
Definition 11.1. Three points x, y, z E Kr(&), not necessarily different, are
said to be collinear if one of the following conditions is fulfilled:
(i ) x +y + z is the intersection cycle of К with some line in P\ defined
over к (that is, each of the points x, y, z turns up equally often as the inter-
section multiplicity of that line at this point).
(ii ) x, y, z lie on some line defined over к which is completely contained in
K.
It is clear that the three-place relation L (x, y, z) : % y, z are collinear’ on
Fr(k) possesses the following two properties:
.. ..- -.:.................42...............
11. 1.1. The relation is symmetric, that is, invariant under all permutations
of Xf y, z.
11, 1.2. For any two points x, у G Kr (k) there exists a point z € Pr(fc) such
that x, y, z are collinear.
If, as in the one-dimensional case, the point z of Property 11.1.2 were
uniquely defined, then the three-place relation L(x, y, z) would give rise to a
symmetric quasigroup structure on Kr(fc) (see Definition 1.1). For the reasons
given in Section 1, this is in general not true. This circumstance suggests the
following definition.
Let, quite generally, P be some set, Z CPX PX Pa three-place relation
which satisfies Properties 11.1.1 and 11.1.2 (with P instead of (&)). Any
equivalence relation R CPX P defines a quotient three-place relation L/R on
the quotient set P/R. This quotient relation also satisfies Properties 11.1.2
and 11.1.2.
Definition 11.2. An equivalence relation R is admissible (with respect to L)
if L/R induces a symmetric quasigroup composition law on P/R (in the sense
of Definition 1.1).
It is clear that the set of admissible equivalence relations is non-empty (e.g.,
A = P X P is admissible).
The following simple result is independent of any additional assumptions
on P or L.
Proposition 11.3, Under the conditions given above on P (and L) there
exists a unique finest admissible (with respect to L) equivalence relation. We
shall call this the universal equivalence relation.
Proof. Let {Rftei be some set of equivalence relations on P. We then define
a new equivalence relation R by
x ~ у (mod A) «=» Vz EI, x ^y (mod R.) .
We shall show that R is admissible if all Rj are. This will prove the proposition
because we can take for Z the non-empty set of all admissible equivalence re-
lations. In fact, let X Y, Z be equivalence classes mod R such that for some
x GX, у G Y, z GZ we have (x, y, z) EL. It is clear that X = ПX^ Y = Л Yif
Z = Л Zz-, where Xit Y-, Zt are the respective equivalence classes mod Az of
x, y, z. Now let x GX, у G Y and (x', y, z) EL.We then have
xr GX, y' G Y x GXp yr G Yj,
(x',y', z)EL z EZ. ,
since all Rj aw admissible. Consequently, zr Ef}^ = Z. The classes X, Y
therefore uniquely define the class Z in the three-place relation L/R, which is
what we wanted to prove.
Remark 11.4. LetRp R2 be admissible equivalence relations, and let
Ei ~ P/R j (z = 1,2). If R j is finer than A2, then the natural map E± £2 is
obviously a morphism of symmetric quasigroups (with composition laws in-
duced by L/Rit i = 1,2). The result of Proposition 11.3 therefore admits a
dual: there exists a ‘largest’ quotient quasigroup ofP; all others are quotients
of this one.
11. 5. The main aim of this chapter is to introduce and study various admis-
sible (with respect to collinearity) equivalence relations on the set Lr(/c). We
shall systematically consider the following admissible relations:
(i) The universal relation (see Sections 13 and 16).
(ii) R-equivalence. Roughly speaking, two points x, у E Yr(k) are R-equiv-
alent if they can be connected by a chain of rational curves which lie com-
pletely in V (see Sections 14 and 15).
In Chapter VI we shall introduce and study in detail:
(iii) Brauer equivalence. To define this, one considers ‘Azumaya—Grothen-
dieck’ algebras over V. Informally speaking, these are sheaves of simple algebras
over V. Each point x E VT(k) then defines a simple algebra A (x), the stalk of
A at x. Two points x, у E Ег(£) are called Brauer equivalent if for all A the
elements of the Brauer group of к defined by A (x) and A{y) are equal. Brauer
equivalence is coarser than R-equivalence; its advantage is that by definition
it can be effectively calculated.
Given some mild restrictions on F, which we do not formulate here, one
gets the following facts:
Let S C Fr (к) X VT (k) be some admissible equivalence relation, E~ VT (k)/S
the corresponding symmetric quasigroup.
Theorem 11.6. E is a CH-quasigroup (cf Definition 1.3).
Now choose some class U € E and construct on E the commutative Moufang
loop with unit element U (see Definition 1.4). Let the dimension of К be
greater than one.
Theorem 11.7. The CML E ~ with composition law XY = U о (X о Y)
satisfies the identity X6 = 1. In particular, the CML has a finite number of
generators if and only if it is finite, and then the number of elements is of the
form 2a3b.
In the case that the base field к is local or global one has some finiteness
theorems. Here is the local result:
Theorem 11.8. Lei [k : Qp] Then the points of (k) ~ Vfk) are par-
titioned into a finite number of classes under universal equivalence.
The global theorem will be proved in Chapter VI, but only for the coarsest
equivalence (i.e., Brauer equivalence).
We give some examples which show that admissible equivalence relations
can quite well be non-trivial.
Example 11.9. Let к = R and V a cubic surface over R such that F(R) con-
sists of two connected components Xo, X| and such that Xo is convex. Then
this partitioning into components is admissible and the CML (Xq , X^) with
zero element Xq is isomorphic to Z2.
Using this, one can for any n > 1 construct an algebraic number field к, а
surface V over k, and an admissible equivalence relation 5 on Kr(/c), such that
Z^ C VX(K)/S. In particular we see that the set E can be arbitrarily large over
number fields. For more details see Example 16.2.
Example 11.10. Let к ~ Q2(0), 1 (this is a non-ramified quadratic ex-
tension of the 2-adic numbers). Let the surface V be given by the equation.
T3 + T3 + T? + в Tl = 0 .
v X «М1 О
Two points x, у EV(k) will be called equivalent if they coincide after reduc-
tion mod 2. In Example 16.3 it is proved that this is an admissible equivalence
relation and that the corresponding CML is isomorphic to Z2 X Z3.
Further examples are contained in Chapter VI.
We now list the main unsolved questions.
Problem 11.11. Can the CMLs E - Vr(k)/S be non-associative (for some
К К 5)?
Problem 11.12. Is there a global finiteness theorem for universal equivalence
and R-equivalence?
Problem 11.13. Is it possible to parametrize an R-equivalence class by a
finite number of parameters; let us say over a field k, [& : Q] < °°?
These and related problems will also be discussed in Chapter VI, in partic-
ular after we have studied the Chatelet example in which all main ideas can
be checked in detail.
Amongst the algebraic—geometric circumstances which influence the struc*
ture of Kr(fc) and its quotients E, the most important is, no doubt, the unira-
tionality of a cubic variety V. The next section is devoted to an elementary
study of this property.
12. Unirationality
We recall the general definitions as regards rational maps of algebraic vari-
eties over a field k.
Let К W be two varieties. We consider the set of pairs (U, </>), where UC V
is a dense open subset of Vf and : U W is some morphism. Two such pairs
(t/p and (I/2, v?2) are equivalent if 1{/1П^2 = </>2 u2 • (The sym-
metry of this relation is obvious, and the transitivity follows from the fact
that W is separable.)
Definition 12.1. An equivalence class/of pairs (17, is called & rational
map of the variety V into the variety W.
Among all morphisms : U IE belonging to a given class there clearly
exists one with a largest domain of definition, the union of all U of the given
class. This open subset of V is called the domain of definition of /
Let f : V~+ W be a rational map, represented by the morphism y : U -> W.
This map is called dominating if the image <p(U) is dense in W. It is clear that
this property does not depend on the choice of the representing morphism.
Lemma 12.2. There exists a natural one-one correspondence between
dominating rational maps f: V -> W and field inclusions f* : A: (IV) -> k( V) of
the rationaifunctions on F, W (these are the local rings in the general points).
Proof. If/ is dominating, then there exist open affine sets V\ С V, C W
and a morphism y : Fj -> PV* representing/such that ^(Fj) C W^. The gen-
eral points of И and W lie in V\ and respectively, and as ^(Fj) is dense
in Wy, the general point of must belong to ^(F[). Consequently, $ induces
an inclusion к (И7) -> k(V)\ it is obvious that this inclusion is independent of
the choice of
Conversely, let there be given an inclusion fc(IV) -> k(V). We choose rings
of finite type A C A; (IV), В C k(V) such that A C B, and such that the quotient
field of A (resp.B) coincides with А/IV) (resp. &(F)). Each element of a fixed
finite system of generators of A is, as a function on IV, defined on some dense
open subset of W. Therefore W and Spec (Л) have a dense open affine subset
in common. By localizing A if necessary, we can assume that = Spec (Л)С IV;
analogously we can arrange things such that Fj = Spec (В) С V. Then the in-
clusion ЛЕВ defines a morphism Fj JVp It is not difficult to check that
different choices of Л and В yield equivalent morphisms.
The constructions of rational maps and field inclusions that we have just
described are inverse to one another; this proves the lemma.
Let/: V -+ W be a dominating rational map. The number [fc(F) :/*(&(IV))]
is called the degree Gif.
Suppose we have two rational maps V 4 IV 4 Z. Then the composition of
/and g can be defined in the obvious way if/is represented by a morphism
ip: W such that <p(U) has a non-empty intersection with the domain of
definition of g. In particular, the composition is defined if/is dominating.
Definition 12.3. The rational map/: V -> IV is called birational if the fol-
lowing equivalent conditions are fulfilled:
(i) / is dominating, and there exists a representing morphism : U -* PV
such that induces an isomorphism of schemes U $(U) E IV.
(ii) f is dominating, and the field inclusion /* : &(JF) &(K) is an isomor-
phism.
We leave it to the reader to verify that conditions (i) and (ii) are equivalent,
using the proof of Lemma 12.2 as a model.
We note that the birational maps f: V -> V form a group which is canoni-
cally isomorphic to the group of isomorphisms of the field k(V) over k. A
birational morphism is any morphism which is birational as a (rational) map.
Definition 12.4. An и-dimensional variety К over the field к is birationally
trivial if one of the following two equivalent conditions is satisfied:
(i) There exists a birational map Рл -> V defined over k.
(ii) The field of rational functions on К is a purely transcendental exten-
sion of the field k.
From the diophantine point of view, this property says that ‘almost all’
points of V which are rational over к are described in a one-one way by n’
independent parameters with values in k. Two examples follow:
Example 12.5. Let К be a cubic hypersurface in P" given by an absolutely
irreducible cubic form. Let x €E К be a singular point which is rational over k.
We call it conical if any line in Pn which passes through x is either completely
contained in V or intersects V only at x. A two-dimensional cone with its
vertex at x and as a base a non-singular cubic curve is, of course, birationally
non-trivial, and the point'x on it is conical.
We show that V is birationally trivial if the point x is not conical. Let Pz?-i
be any hyperplane which does not pass through x (see Fig. II.l). The function
/: Рп“1 V which maps a sufficiently general point G p«-l onto the point
(different fromx) of intersection of the line through x,y with V is birational.
Indeed, let us write, the equation of K in some affine coordinate system in
Pw withx as the origin and P^”1 as the hyperplane at infinity:
^(т1,...,тй) + с(т1,...>т?г) = о.
Here (Tp ... , Tn) is at the same time a homogeneous coordinate system on
P””1; q is a quadratic form and c a cubic form. We have that q Ф 0 because x
is non-conical; the linear part vanishes because x is a singular point.
Fig. П. 1.
We denote by U C P”"1 the complement in the hyperplane of q - 0, c = 0;
and we find that for a point у = ,..., G U the line through x, у inter-
sects V at the point
f(y) = ,\tn)^x = (0,0,,0),
-c(rp...,Q •
Therefore/induces an isomorphism/: U % ({c = 0}U {q = 0}).The
latter complement is open and dense in V, because V is irreducible.
Example 12.6. Let V С P? be a cubic surface on which there are two non-
intersecting lines defined over k. Let ip, ф : P1 -> V be the embeddings of
these two lines in V. Construct the map о ф : P1 X P1 -* V, which maps a
point (x, y) G P1 X P1 onto the third point of the intersection of V with the
line passing through and This map is defined on a non-empty open
subset of P1 X P1; otherwise V would contain a plane (consisting of the lines
through one of the points^(x) and intersecting ^(P1)). Moreover, this map is
one-one on a dense subset of V, because one can pass through any point of P3
outside ^(P1 *) and ^(P1) exactly one line which intersects both^P1) and
^(P1). By translating these geometric considerations into algebraic language one
easily convinces oneself that the map ф is birational.
A concrete example is the surfaceXq + x3 + x2 f x3 = 0 over a field which
contains a primitive cubic root of unity 6. This surface contains for example
the non-intersecting lines x0 + Xj = x2 + x3 =0 and x0 + x3 =x^ + 0x2 = 0.
12.7. The interest of the following theory is connected with the fact that
cubic hypersurfaces are, generally speaking, not birationally trivial, even if
they have ‘many’ rational points. We restrict ourselves to non-singular varieties.
For these the following two facts are well known:
(i) dim V~ 2. If the field к is algebraically closed, then V is birationally
trivial, because it is then always possible to find a pair of non-intersecting
lines on К Over non-closed fields there exist many birationally non-trivial
cubic surfaces. The problem of their birational classification and of the struc-
ture of the group of birational maps has been thoroughly studied, although it
is not finished yet. The known results will be presented in Chapters IV and V.
An example of a non-trivial surface is
x3 + x3 + x| + ax$ - 0 , a (A:*)3 .
(ii) dim 3. The question of birational triviality is an unsolved classical
problem already in the case к ~ C and dim V - 3.1 Over non-closed fields one
can construct birationally non-trivial V; however, no general results on the
classification are known to the author.
It is, however, extremely important for the following that most cubic
hypersurfaces have the following property, which is weaker than birational
triviality:
Definition 12.8. An n-dimensional variety V qnw a field к is called unira-
tional if one of the two following equivalent conditions is satisfied:
(i) There exists a rational map of finite degree P/7 V defined over k.
(ii) There exists a rational map Pm -> V defined over к which is domi-
nating (that is, it induces an inclusion of function fields A;(K) -> fc(Pw)).
1 Clemens and Griffiths have recently shown that smooth three-dimensional cubic hyper-
surfaces over C are birationally non-trivial.
Proof of the equivalence, (i) => (ii) is clear.
Conversely, let there be given a map f: Pm -> V with property (ii). There
exists a dense open set U C such that/is defined at each point of U and
such that the dimension of the fibre is constant and equal to m - n on 77.
Take a rational point x G U and construct through it a linear subvariety
рл q pm which is defined over к and which is transversal to the fibre/^(/(x)).
The restriction of /to Pn is an epimorphism in the general point and has there-
fore finite degree. This concludes the proof.
If the variety V is unirational, then the set of its ^-rational points is dense
in the Zariski topology. Of course, the converse is not true, even for non-singu-
lar cubic curves.
The main result of this section, however, establishes this property for a
large class of cubic hypersurfaces of dimension > 2. We need the following
definition.
Definition 12.9. Let TC Pn be a cubic hypersurface. A geometric point x
of V is called a point of general type if the following conditions are satisfied:
(i) x is a non-singular point of К
(ii) The intersection C(x) of the tangent hyperplane at x with V is geo-
metrically irreducible and reduced.
(iii) The singular point x G C(x) is not conical on C(x).
Example 12.10. If dim V = 2, then a non-singular point x is not of general
type only if C(x) is geometrically reducible. One of the components is then
a line and passes through x.
A schematic picture of all possible shapes of C(x) is shown in Fig. II .2.
Here x is of general type in the cases (a), (a^ and (b). In case (e),x is a conical
point on C(x). In case (a) there are two tangent lines to C(x) at x, defined
(b) (c)
Fig. II. 2.
(d) (e)
over the base field, in case (b) they are defined over some quadratic extension
and in case (zzj) the two tangent lines coincide. Pictures (d) and (e) do not reflect
the action of the Galois group on the geometric components; it may be non-
trivial.
Theorem 12.11. Let V CPZI be a cubic hypersurface of dimension > 2
which contains a k-pointx of general type. Then V is unirational and we have
in particular that the set V(k) is dense in the Zariski topology.
Proof. First we remark that the points of general type form an open set
on F. In fact, each of the conditions (i), (ii) and (iii) is open: (ii) because it is
equivalent to the requirement that the tangent hyperplane at x does not con-
tain any linear subvariety Рл~2 c V\ and (iii) because the condition ‘conical’
can obviously be expressed by means of the matrix of values of the second
derivatives of the equation of V.
It now follows from the conditions of the theorem that the set of points
of general type on V is dense. The variety C(x) is birationally trivial in virtue
of Example 12.5: the Appoints of general type on V, which lie in C(x), form
a dense open subset U C C(x) and for each A:-point у G U we can form the
birationally trivial variety С(^) on V. Thus we obtain a system of birationally
trivial varieties C(j^) on F, parametrized by the points of the birationally triv-
ial variety U, We need to show that this system forms a birationally trivial va-
riety and that its natural map into V is epimorphic in the general point.
Leaving the details to the reader, we give a sketch of one of the possible trivi-
Fig. П. 3.
Fig. II.3). Choose a hyperplane Нл-1 С Рл which is not tangent to V at x.
Let DM_1(x) be the tangent hyperplane at x, and let CHn~^ be a hyper-
plane in such that -й^«“1(х). Finally choose a point
z € H77-1 \ H7*’2. The hyperplane H71”2 is birationally mapped on the fibre
C(y) for almost ally in the following way. We project a-point v € Ни~2 from
z onto the hyperplane DM~I(y) П H77-1 (inside H77-1). After that, draw the
line joining the point thus obtained toy (within D77™1^)); intersect this line
with К and let fy(p) denote the intersection point different fromy. The map
t/X H”-2 -> К: (y, v)t-+fy(y)
is rational. It is epimorphic in the general point because otherwise its image
would be contained in C(xf But then we would have C(y) = C(x) for ally,
which is impossible because every pointy is singular in C(y), but (for a dense
set) non-singular in C(x). This proves the theorem.
Remark 12.12. A two-dimensional cone has no points of general type.
I do not know a complete description of hypersurfaces with such a prop-
erty. One can show that there are always geometrical points of general type on
a non-singular hypersurface, so that a non-singular hypersurface of dimension
> 2 over an algebraically closed field is always unirational.
To conclude this section we state the following simple result.
Proposition 12.13.£e?x G V be a non-singular к-point of the cubic hyper-
surface Vt and C(x) the intersection with V of the tangent hyperplane at x.
There exists a unique birational map over k, tx\ V defined outside C(x),
such that the points (x, y, Zx(y)) lie on one line for у G C(x).
[We shall study the precise structure of tx later (see Section 21.2).]
Proof. Choose a homogeneous coordinate system (Tq, ..., Гл) in which
x = (1, 0,. .. , 0) and Ty = 0 is the equation of the tangent hyperplane to V
at x. The equation of V in affine coordinates will then be
т ГТП
i _
i г • • • > rji
0
and in homogeneous coordinates,
Tj T%+ TQ q{Tv c(T},.. .,Tn) = 0----- (II.l)
The line through the points (1, 0,..., 0), ,..., Z^) G Khas the param-
eter representation 7q = X + дZq, 7) = pt-, i > 1. Substituting this in formula
(II.l) we find
Хд(Ц +м(2г0?! +<7(Гр ... ,Г„)) = 0 .
On D+(T1) we can take
M ty ,
~~ Vi + > • * • > ’
from which
^0 ~ V1 + > * • * >tn) ,
and therefore
^(1,0,..., 0)^0’ (rofl + ^P ~ Vl’ ”* ’ ~^п)' -
This is a morphism on D^T^).
13. Universal equivalence
Let К be a fixed cubic hypersurface over k, Vx(k) the set of its non-singular
^-points, andL С Vr(k) X У(к) X Уг(к) the collinearity relation defined in
Definition 11.1. We start here the study of equivalence relations on Vx(k)
which are admissible with respect to L in the sense of Definition 11.2.
We shall say that the points x, у G Pr(k) are in general position if x Фу
and the line through x and у is not contained in V and is not tangent to V,
In this case, there exists a unique point z G Kr(fc) such that (x, yt z) are col-
linear, and z - tx(y) ~ ty{x\ From now on we shall write z "x cj’ = у о x in
this case. This notation is justified by the following: if У, У, Z are the classes
of x, y, z with respect to some admissible equivalence relation, then Z = X о У
in the sense of the composition law induced by L.
Theorem 13.1. Let V be a cubic hypersurface of dimension > 2 having a
к-point of general type, let S be some admissible equivalence relation on
and E = Vr(k)/S the corresponding symmetric quasigroup. Then the following
statements hold'.
(i) Every equivalence class is dense in the Zariski topology. If к is alge-
braically closed, then all points of VT(k} are equivalent to one another
(ii) E is a CH-quasigroup.
(iii) Let B(V) be the group ofbirational maps of Vgenerated by the maps
tx for all x G Уг(к). Then there exists an epimorphism of groups
B(V)^T(E).tx^tx,
where X is the equivalence class of x.
(We recall that t: E E is the map ^(У) ~ Хь У.)
Proof, (i). We start with the following important remark:
Let x G Lr(A:) be any point, C(x) the intersection of V with the tangent
hyperplane at x. Then all the к-points lying on C(x), except possibly x, be-
long to the same equivalence class. In fact, if G C(x),у Фх, the line con-
taining x, у is rational over к and it is either tangent to V at the point x or it
is completely contained in V. The points (x, x, y) are therefore collinear and
that means that у G X о X, where X is the equivalence class of x.
Applying these considerations to the points of general type у G C(x) \ {x},
where x is some fixed point of general type, we get that all points of the
set U^(C( j) (jp)) (k) are in one and the same class (X о X) о (X о X). The
proof of Theorem 12.11 shows that this set is dense in the Zariski topology. At
least one ^-equivalence class У is therefore dense.
Let X be any other equivalence class. We shall show that it is also dense.
Choose a point x0 G X; since it is non-singular, the points ^0(j) are defined
for a dense subset of points у G У, they belong to the same equivalence class
X о У and form a dense set. Now let G У be a point such that j'q and x0
are in general position. Then again there exists a dense set of points of the
form tyQ <x0(y) belonging to the same equivalence class, viz. the class X, be-
cause it contains ^0^x0(To) = x0* — -------- '
Finally, let к be algebraically closed. Then for each point x E YT (k) there
exists a line which either has a threefold intersection with V at x or is in K,
so that for any equivalence class X and A' Gio X we have x EX о X **X=X о X,
and all points of C(x) are equivalent. Therefore all points of Upecfx)
are equivalent. But this set is obviously thick, that is, it contains all ^-points of
some open Zariski dense set. The arguments of the preceding paragraph then
show that any equivalence class is thick and, as a consequence, all classes coincide.
(ii) . To verify that E is a CH-quasigroup (Definition 1.3), one must establish
that any three classes X Y, ZEE generate an Abelian subquasigroup. Since all
classes are dense, one can find points x EX,y EY, z EZ such that the plane
through x, y, z intersects V in an absolutely irreducible cubic curve (Bertini’s
theorem) on which x, y, z are non-singular points. The collinearity relation in-
duces the structure of an Abelian symmetric quasigroup on the й-points of this
plane section of V (Example 2.2). The subquasigroup of E generated by X Y, Z
is a quotient of this Abelian quasigroup and therefore is also Abelian (Corollary
2.3). Hence £ is a CH-quasigroup.
(iii) . To construct a homomorphism B(V) T(E), we first define an action
of B(Y) onE. Let s = tXy... tXn EB(Y), Y EE, Since' Y is dence, there exists
a point у E У such that for all z, 1 <z< n, the points x2- and tXi+i ... tXn(y)
are in general position. We then define
s(Y) = class tXi... tXn(y).
To show that this is well defined we must verify that the definition does not
depend on the choice of y, or on the factorization s= tY ... tY , and that
S1(S2(Y)) = (S1s2)(y\
The independence of the choice of у is verified by induction on i. The class
of tx (y) does not depend on the choice of у because the points (xn, y, tx (y))
are collinear and the classes of xn and у are fixed. The step from i + 1 to i
goes similarly. The independence of the factorization of s can be shown as
follows: we can choose a pointy G Yin the class Y such that for two given
factorizationss = tY ...tY ... t. we have that for all i ~ 1,..., n and
j = 1,...,m the pointsx.-, E ... tY (y)andzh t„ ... t,
position. It is then obvious that tx ... tx (y) = tz ...tz (y) = s(y). Because
the class of the first and the second point does not depend on the choice ofy,
(y) are in general
we obtain that the class s(T) is well defined. A completely similar argument
shows that (s2(^)) ~ (sjS2)
Thereby we have defined a leftaction of the group В (И) on E for which
tx (for all x G Kr(&)) acts as t%, where X is the class of x. This defines the de-
sired epimorphism В (К) T(E) and completes the proof.
Theorem 13.2. Let the conditions of Theorem 13.1 be satisfied and intro-
duce on E a CML structure by means of the composition law XY = Uo (Xo У),
where U^E is some fixed class. Then the relation X& = 1 holds in E.
Proof. We show that the identity 0(X) = 02(X) is true in E, where
0(X) = X о X. In fact, let X G E be any class, x E X a point of general type.
We choose on C(x) two ^-pointsj' and z in general position. Then we clearly
have у о z G C(x), у о z Ф x. On the other hand, by the remark in the first
paragraph of the proof of Theorem 13.1,y, z} у о z EXoX. This proves our
identity. (The preceding arguments are due to A. Bel’skii; in [9], I have
proved this result in a more complicated manner and only for R-equivalence.)
Let us now put this identity in terms of the CML multiplication. In virtue of
5.1.2, we have X о Y - 0(1) X~x У’1, hence 0(JQ = 0(i)X~2 and 02(X) =
= 0(1)“!X4. Consequently, X^ = 0(1)2. This is true for allX, in particular for
X = 1 = U, therefore X6 = 1. This proves the theorem.
Corollary 13.3. E is the direct product of an Abelian group of exponent 2
and a CML of exponent 3.
In fact, it follows from Moufang’s theorem (Theorem 4.1) that the map
xf->x” is a CML morphism for every integers. In particular, let E2, E^ CE be the
subloops consisting of squares and cubes of elements of E, respectively. Then the
map E ~*E2 X : x >-» (x2, x3) is a CML morphism. Its kernel is trivial, and
(u, v) им-1 is the inverse map. Finally,E2 is a CML of exponent 3, and is
an Abelian group of exponent 2. (We can of course prove a general theorem
on the decomposition of a torsion CML into p-primary components; for p =#3
these components are Abelian groups by Theorem 1.8.)
Corollary 13.4. If E has a finite number of generators, then it is finite and
consists of 2a 3^ elements.
Below, it will be proved that this result can be applied to complete locally com-
pact fields.
Sometimes one can show that the 2 or 3 component is absent. First of all,
by refining the considerations used for Theorem 13,l(i) we obtain the fol-
lowing useful result. ;
Proposition 13.5. We assume that any quadratic form in > r variables has
a non-trivial root in the field k, and that the conditions of Theorem 13.1 are
satisfied. Then the CH-quasigroup E is distributive if the dimension of V is
larger than or equal to r. In other words, the identity X - XoXholds in E for
all.Xand the identity X$ = 1 is satisfied in the CML with composition law
XY - U о (X о Y). In particular, a quasigroup E with this property and which
has a finite number of generators is finite, and its number of elements is of
the form 3K.
Proof. Let V С Pn+1, x E Fr(fc) an arbitrary point. Choose in Pn+1 a coor-
dinate system (Tq, ..., Тл+1) such that x = (1,0,..., 0) and such that 7^ = 0
is the equation of the tangent hyperplane at x. The equation of C(x) will
then have the form (in the hyperplane Ty ~ 0 with affine coordinates
ЗД, • • • >
(Гр rp \ yrp m 4
12 л+1\ Г 2 7«+1|_л
гр гр I , • • • гр f 0 J
J0 70 7 Л7о 70Z
where q is some quadratic and c some cubic form. If n > r, then the equation
q = 0 has a non-trivial solution in k. This solution defines a line in C(x) which
is defined over к and either has a threefold intersection with V at x (when
с Ф 0), or lies completely in V(when c - 0). In both cases we obtain XоX ~X,
where X is the equivalence class of x.
The remaining statements about the structure of E follow from the results
of Chapter I; see in particular 5.1.4, and Theorems 1.9 and 1.10.
13.6. This result can be applied to a rather large class of fields; for example:
(i) If к C F? (the closure of the prime field of p elements), then r - 3.
(ii) If A: is a non-Archimedean local field, then r = 5.
(iii) If к is a purely imaginary extension of the rational numbers, then r = 5.
(iv) If A: is a Q-field, then r = 2Z+ 1;
(v) If к has no non-trivial quadratic extensions, then r = 2.
In addition we remark that the simplest example of a geometric CH-quasi-
group with the relation X о X = X is formed by the inflection points of a com-
plex non-singular cubic curve;~TKefe^fHMnF^cff points, and the corre-
sponding CML is isomorphic to Z3 X Z3. The construction of Example 16.3
is based on this idea.
Bel’skii has shown that the 3-component is absent if V contains a line:
Proposition 13.7. We assume that the conditions of Theorem 13.1 hold and
that there is a к-line D С V of which not all points are singular on V. Then, if
we designate by U the class of points ofD(k) П Lr(A;), we have XoX= U for
all X^E. In particular, the CML E with unit element U is an Abelian group
of exponent 2.
Proof. Let x G X be a point of general type, x A Construct a plane P2
through x and D, We can assume that the intersection P2 П V decomposes
into D and a geometrically irreducible conic Q. Construct in P2 the tangent
P1 to Q at the point x; let the intersection of P1 with/) be y. It is clear that
у is non-singular on K; moreover, 2x + у is the intersection cycle of V with P1.
Therefore X оX ~ U, as we desired to prove.
Example 13.8.1. dim V - 2. There are 27 lines on V ® к (see Chapter IV),
but it can happen that none of them is defined over k.
Example 13.8.2. dim V = 3. For each sufficiently general point on V ® к,
there are six lines passing through it; in terms of the notation of the proof of
Proposition 13.5, their union is given by the system
If V is smooth, then the set of all lines on V ® к is parametrized by an ex-
tremely interesting surface. Its Albanese variety is 5-dimensional; cf. Bombieri
and Swinnerton-Dyer [1]. I do not know whether the set of Ariines on V can
be empty or infinite, for instance for a number field or a local field. This is
an interesting diophantine problem.
Example 13.8.3. dim V = n > 4. If the field к is such that any system of
equations of the form (II.2) is solvable, then E is trivial: there is no 2-com-
ponent according to Proposition 13.5, and no 3-component according to
Proposition 13.7. This argument can be applied to Ц-fields (dim 2l + 3Z),
and probably also to local fields and purely imaginary global fields for dim V
sufficiently high.
13.9. To conclude this section, we point out an algebraic—geometric descrip-
tion of universal equivalence on (cf. Proposition 11.3). This description
can be useful for a study of E with the methods of Chapter V.
Let В0(К) denote the normal subgroup of 7?(K) generated by elements of
the form txtytztxdy> tzt> where (x, y, z) and (x',y',z') run through all pos-
sible triples of collinear points of Ут(/с). Define x ~y (mod U) if txf, е50(Ю-
It is clear that U is an equivalence relation on Lr(fc).
Theorem 13.10. Under the conditions of Theorem 13.1, U is the universal
admissible equivalence relation.
Proof. First of all, we show that U is admissible. Let (x, y, z) be collinear;
one must verify that the £7-class of z is defined by the £7-classes of x andy.
It suffices to verify that if x ~ x (mod U) and if (x1, y, z‘) are collinear,
then z ~ z (mod £7). In fact, x ~x (mod £7) if and only if txtx> G/?q(K).
Moreover, tx ty tz tx,ty tz> G/?0(K) by definition. Because Bq(L) is a normal
subgroup ofB(T)} it follows that tztz, G2?0(K), that is, z ~ z (mod £7).
Now let 5 be any admissible equivalence relation. We shall show that
x ~ у (mod £7) impliesx ~у (mod 5). Let/? = VfE)\S and let : B(V)->T(E)
be the homomorphism constructed in Theorem 13.1. Using the fact that E is
a CH-quasigroup and taking into account Theorem 3.1 (ii), for any collinear
triple of points (x, y, z) and (x', y, z ) we obtain
= zX*yzXoyzx'^Х'оУ “ (ryoy)2 = 1 •
Here X, Yf ... are the classes of x, y,... mod 5. As a consequence, 7?q(K) C
C Ker <p. Therefore, we have
x~y(mod U)e>txty GB^(V)^<p(txty) = txty~ \ &tx = ty ффХ= У.
This concludes the proof.
Corollary 13.11. Under the conditions of Theorem 13.1, let E ~ V/k)IU
be the universal CH-quasigroup of classes of points on V. Then there is a ca-
nonical isomorphism T(E) "B^V^/Bq(K).
We remark that the group can also be defined for hypersur-
faces over finite base fields. I do not know whether this group is necessarily
trivial in this case, at least when the number of elements of the field is suffi-
ciently large.
14. R-equivalence: the basic properties
Following Chevalley, we call a variety which is irreducible over к special if
each of its points possesses an affine neighbourhood which is isomorphic to an
open subset of some affine space Spec fc[7|,..., Tn ]. We shall say that a rational
map of varieties f\U-+V covers a point x E V(k) if there exists a point
и e U(k) such that /is defined at и and/(w) = x.
Definition 14.1. ^-equivalence on K(£)is the weakest equivalence relation for
which any two points covered by one and the same rational map of a special
variety U into V are in the same equivalence class.
An amusing example is the following: Let Fbe an irreducible cubic curve
with a double point x. If the two tangent lines to V at x are both defined
over the base field, then all points of V(k) are R-equivalent. Otherwise, V(k)
splits into two classes, namely, {x}and the class of all other points. It is this
very phenomenon, however elementary, which is responsible for the existence
of a 2-component in our CML s (cf. Proposition 13.5).
Lemma 14.2. Two points x,y£ V(k) are ^-equivalent if and only if there
exists a finite sequence of points x = Xj, x2,..., xr =y, and morphisms
f.. pl _> v, i = 1,..., r— 1, such that /z- covers xif xz+1.
Proof. It is clear that the condition is sufficient. To prove that it is also
necessary, we remark that it follows from the definition of R-equivalence that
there exist points x = x j, x2,... , xr = y, morphisms of special varieties
fi: U^V and points ufl, ui2 G Ufk) such that //(wfl) = x'it f^Uff) ~ x'-+1.
If we could join the points u^ and u^ in Ц- by a rational curve on which u^
and u^2 are non-singular, then everything would have been proved. But also
sufficient is a weaker fact, which can be verified easily: and can'be
joined by two rational curves which intersect in an intermediate point v. In
fact, choose open neighbourhoods of the points wzl and which are isomor-
phic to open subsets of an affine space, take a £-point v in their intersection
and join to v by a line within the first neighbourhood and to v by a line
within the second neighbourhood (the ‘lines’ can of course be incomplete).
This proves the lemma.
Just as Definition 14.1, this lemma can of course be applied to any variety V.
We now return to cubic hypersurfaces.
Theorem 14.3. Let V be a unirational cubic hypersurface over k, Then R-
equivalence is admissible on Kr(fc).
First we verify a necessary condition for admissibility (cf. the proof of
Theorem 13.1 (i)).
Lemma 14.3.1. Under the conditions of Theorem 14.3, let x G Kr(fc) be
any non-singular point, C(x) the intersection of V with the tangent hyperplane
at x. Then all к-points on C(x), except possibly x, are ^-equivalent.
Proof. We distinguish several cases depending on the character of the degen-
eracy of C(x):
(i) C(x) is geometrically irreducible.
(a) x is a conical point on C(x). In this case (Example 12.5), any line
joining x with the pointy т^х, у G C(x) (k), is completely contained in C(x).
Therefore у ~x (mod R), that is, all points of C(x) (k) are R-equivalent.
(b)x is not a conical point on C(x). The construction of Example 12.5
yields a rational map of a special variety into C(x) which covers all points of
C(x) except possibly x and points у Ф x for which the line joining у to x
is completely contained in V. Further, in the notation of Example 12.5, such
points correspond to a solution of the system q - c - 0 in k. The ^-points on q =0
are dense because к is an infinite field, and in particular there are solutions
of the equations q - 0, с Ф 0 because C(x) is geometrically irreducible. This
shows that the point x is also covered by the map of Example 12.5 and
that it is therefore R-equivalent with those points of C(x) for which the
line joining them to x does not lie completely in V. As a consequence, all
points on C(x) are R-equivalent-also Tn-thiS" case.
(ii) C(x) is geometrically reducible. Then x lies on the intersection of two
or three irreducible components of C(x) ® k.
(a) C(x) ® к splits into a linear space and an irreducible quadric. Then the
splitting already occurs over k, and both sets of non-singular points of both
varieties are special. It is easy to see that the singular points on the quadric
are also R-equivalent.
(b) C(x) ® к splits into linear spaces. If this splitting arises from one over
the base field, everything is clear. Otherwise, the Galois group permutes the
components, and the fc-points lie on the intersection of all conjugate compo-
nents, which is linear. This concludes the proof.
In addition to this we need two technical results:
Lemma 14.3.2. Let Wbea normal curve, /: W -> К some morphism,
x G V(k)and /(IV) C(x). Then the morphism txf: W -> V is defined as the
composition of the morphism IV X V and the birational map V. If
f(w) G C(x) for some point w G W(k), then also (tx f) (w) G C(x).
Proof. The significance of the lemma lies in the fact that we cannot calcu-
late (txf) (w) as tx(f(w)) since f(w) does not belong, generally speaking, to the
domain of definition of tx. However, the proof is truly easy. If (txf)(w) =
~y t C(x), then (tx(txf))(w) - tx(y) Ф C(x), because V x C(x) belongs to
the domain of definition of tx and is invariant with respect to tx (Proposition
12.13). But tx(tx f) = /, which contradicts the condition/(w) G C(x).
Lemma 14.3.3. Let fa : Ц- -> V (i = 1, 2) be rational maps of special varieties
into the unirational cubic hypersurface V and suppose that f covers the
points Xj,y G V(k). Then there exist rational maps f] such that U[ is
a special variety and
(i) f- covers the points xif у (i = 1,2);
(ii) the set of points of V(k) covered by both f{ and f^ is dense.
Proof. The lemma states that if x^ ~y ~ x2 (mod R) and if each pair (x^y),
(у, x2) is covered by one and the same map, then by changing the maps,у can
vary over a dense set.
For the proof we consider some rational mapg : Pr V with a dense image.
We can find a point a G Vx(k) with the following properties:
(l)g covers a\
(2) a is in general position with respect to Xp x2, y.
Let Xy = ffa^y a = g(b). We take Uff - Ut X Pr,
f'i = (tafi)og:UiXPr^V.
The rational map ta is defined here in virtue of condition (2), and the
composition (tafy og is defined because the image ofg is dense. Moreover,
f'Xuj, b) = tf. (и-) о g(b) = (a о x.) о a = x.,
£ » 4* it I t
b) = t f.{u^ O g(b) = (aoy)oa=y.
Finally, the domain of definition of the restriction of f- to и\ X Pr is non-
empty and this restriction coincides with the rational map
the image of which is dense. The points covered by this map are covered at the
same time by/j and /2, which proves the lemma.
14.3.4. Proof of Theorem 14.3. Let (x, y,z) and (x',y,zf) be collinear triples
of points of Vr(k) and let x ~ x'(mod A). If suffices to verify that then
z ~z' (mod R). According to Lemma 14.2, one can restrict oneself to the
case where there exists a morphism /: P1 -> V covering x and x.
If у is in general position with respect to x and x', then it is clear that the
morphism ty f: P1 V covers z and z', so that this case is trivial. If у is in
general position neither with x nor with x, then according to Lemmas 14.2
and 14.3.3, one can join x to x by a chain of rational curves for which
the intermediate intersection points are in general position with respect to y.
It is therefore sufficient to treat the case where у is in general position with
x , but not with x. In this connection one can even assume that x G C(X);
otherwise, ty would be defined in x and the same considerations as above
would give the desired result.
Let/(a) =x, f(u) = x'. Then (^/)(uz) = ty(f(u))- ty(x')=z\ since .y and
x are in general position. On the other hand, z ~ (tyf) (u) (mod A). The point
z"= (Lp/)(u) lies on C(y) according to Lemma 14.3.2. The point z also lies on
C?(j) because x ^C(y) and (x, jyz) are collinear. From Lemma 14.3.1, one
immediately obtains that the points z and z" (and therefore z and z') are R-
equivalent in all cases, except maybe in the case that z ~y, z” Фу and no lines
exist which are defined over к and either are tangent to V threefold at у or
contain у and are completely in V. However, this is impossible. In fact, if
these conditions are fulfilled, thenx фу.(because (x, y, y) are collinear) and
the line which passes through x, у does not lie in V and is tangent to V two-
fold at y. It follows from this that у is the only specialisation of (tyf)(w)
(where w is the general point of IV) extending the specialisation w -> u. There-
fore z" =y, contradicting the assumption.
Remark 14.4. R-equivalence is, in contrast with universal equivalence, bira-
tionally invariant in the following sense: Let Fp be unirational cubic hyper-
surfaces andFj F2 some birational map. Then/induces г set isomorpnism
5 ^2д-(А;)/А (in general, the composition laws do not coincide!).
This follows easily from the fact that the R-equivalence classes are dense; we
leave the details to the reader.
In Chapter V it will be proven that for an important class of cubic surfaces
also the quasigroup V(k)/R is birationally invariant and so is even Vx(k)IU
(where U is universal equivalence). This is a very subtle fact.
Remark 14.5. The only class of varieties for which the calculation of R-
equivalence succeeds completely in explicit form are the Chatelet surfaces.
This example is analysed in Chapter VI because for the proof one. must also
use Brauer equivalence.
15. R-equivalence and quadratic extensions
In this section we study the behaviour of E ~ Fr(k)/R under a quadratic
extension of the base field. Let А Э к be an extension of k. Then clearly
there exists a morphism of CH-quasigroups i = z/^: = VT(k)/R -> Eg =
~ VT(K)/R which maps the class of a point x G Vr(k) onto its class in Fr(A).
We denote by X& the image of X ^E% in EK. It will be shown that when
К Э к is a separable quadratic extension, there exists a ‘norm’ map in the
opposite direction (not necessarily a morphism):
Proposition 15.1. If [A": &] =2 and К is separable over k, then there exists
a map of sets N ~ : Eg -> Ek such that N о i(X) = fi(X) for all X£Ek.
(RecaB that 3(X) = X о X.)
Corollary 15.Ll.Ler the conditions of Proposition 15.1 be satisfied and let
X, Y ^Ek. Then i(X) = i(Y)=$ (3(X) = 3(Y). In other words, the kernel of the
CML morphism i consists only of elements of order two.
. In fact, i(X) = i(Y) ^Noi{X} = No i(Y). In terms of CML’s: 3(1)X"2 =
= 3(1) Y~2 =>X2 ~ Y2. One can obtain from this another proof of the identity
X6 - 1 in the CML E = V(k)lR (cf. Theorem 13.2):
Lemma 15.1.2. For all X^Ek we have 3(X) = 32(X)-
Proof. We show that there exists for eachX EEk a separable quadratic ex-
tension КЭк such that i(X) = 30(X)) = z(3(X)). The result will then follow
from Corollary 15.1.1.
Indeed, let x G V(k) be some point, and X its A-class. We have already noted
that (C(x)\ {x}) (ft) C (3(X). Moreover, it can be seen from the proof of Lemma
14.3.1 that the point x does not belong to this class only if C(x) is geometri-
cally irreducible and if the quadratic tangent cone to C(x) at x has no ^-points
except x. But then it surely acquires points in some quadratic extension of k,
which is separable, at least if к is perfect or if char (k) 2. This proves what
we wanted.
15.1.3. Proof of Proposition 15.1. We construct the map in the fol-
lowing way: for every class X E Eg choose a point x € X such that x, its con-
jugate over k, is in general position with respect to x. Then x о x G V(kf We
put Ng/k(X) - x о x (mod A) GEk.
We have to prove three statements: (1) it is always possible to choose an
x with the indicated property; (2) the class of x о x does not depend on the
choice of x; (3) if X = z(Y), Y CEk, then Ng/k(X) - 3( Y). In the proof
of the second and third statement we essentially use the fact that we work
with R-equivalence. I do not know whether the analogous facts hold for uni-
versal equivalence, and I see no reason why they should.
(i) The choice of x. We denote by RKfc the functor “restriction to the
base field” of Weil [2]. We recall'that it associates with any quasiprojective
variety W defined over A: a variety^^(HA) defined over к with the following
property:
Let X be an arbitrary variety over k\ then there are defined isomorphisms
which are canonical and functorial in X\
Нот^(А®^А,и/)^Нот^(А,А^д(И/)).
If we apply this in particular to X - Spec (k), we find that the А-points of the
variety W are in one-one correspondence with the к-points of Now
let W= V ® K, where V is defined over k. Taking X ~ R we obtain
Hom* (RK/k(W) ®kK, V ®kK) = Homk(RK/k(W\RK/k(W)-).
Every automorphism of the extension К/к acts on the set on the left by means
of the second factors of RК and V К and that means that it
also acts on the set on the right. In particular, let К/к be normal and let
G = Gal (K/kf Then there is defined a map G -> Нопд (АА^д(И/)),
s^s(id), where id is the identity map of А^д(И^). One easily sees that this
map defines an action of G on А^д(К') for which the induced action on the k-
points of the variety А^д(РЕ) coincides with the action of G on the Appoints
of W=7®*A.
We now apply these general observations to the case where И is a cubic hyper-
surface over к, К Э к is a quadratic Galois extension, and V = А^д A).
Identifying 7(A) with 7(&), we first of all obtain that the set
{(x, x)l (x,x) € V(k) X 7(fc)} is closed (it is the graph of the conjugation mor-
phism). Moreover, the set of pairs (x, y) G V(k) X V(k) such that the corre-
sponding points in 7(A) are not in general position is also closed. It follows*
from this that the set S of points x G 7(A) = V(k) for which (x, x) are not
in general position on V is closed in V(k). It remains to be shown that 5 does ,
not coincide with all of 7(A). Then, as V is unirational, 5 cannot completely
contain any R-equivalence class over A because these classes are dense in V
and hence also in V,
To find points outside S, we intersect V with a sufficiently general 3-di-
mensional projective space, defined over k, and which passes through some
non-singular А-point of V. This intersection is then an irreducible cubic A-sur-
face F on which the point x is non-singular and of general type. Consequently,
all points in some neighbourhood of x are of general type. Therefore, there
is only a finite number of lines on F (their union consists of the points not
of general type). Further, the set F(K) \ F(k) is dense in F (for instance be-
cause the set Cp(x) (K) \ Cp(x) (k) is dense in CF(x) for any point x G F(k)
which is of general type on F). One can therefore find a point x EF(K)^ F(k)
which does not lie on one of the lines on F. Then x Ф x, and the line joining
x and x intersects F(and V) in exactly one more point, which is defined over
k, and therefore different from x and x. It follows that x and x are in general
position, so that x S.
(ii)Ng/k is well defined. Let XE V^K^/R. We choose two pointsx, у EX
such that (x, x) and (y, y) are in general position; we shall show that
x ox ~ у о у (mod F) over k.
According to Lemma 14.2, there exists a sequence of points xz- G V(k\
x =x0, Xp . .. , xr+1 =y such that xz«, xM are covered by a A-morphism
%О)=х., f^=xi+1.
In virtue of Lemma 14.3.3, the points xv . . . , xr can be varied over an every-
where dense set. Therefore, according to the results of (i), we can assume that
хг-, xz- are in general position for all i - 1,..., r. We let the non-trivial automor-
phism of the field K/k act on ® К and V ® К by means of the second
factor. This defines the morphisms We set
and by definition we have for all t E p!(fc),
In particular, gz-(0) = xz ox? g;(°°) = xM ox • It follows from this that the
points x о x = Xq о x0 and у о у = xr+1 о xr+1 are R-equivalent over k.
(iii) The identity Ngо ik^ = /3, Let x EXE VT(k)/R. Again applying Lemma
14.3.3 .and the results of (i), we can find a point у E i(X) and a ЛГ-morphism
К V® К such that у, у are in general position and such that
/(0) = x, /(°°) =y. The morphism g =fo f : P| -> V defined as above covers
g(0) and у c y. It is therefore sufficient to verify that g(0) G /3(X).
Let t E (k(f)) be a generic geometric point iii the sense of A. Weil. In
the space in which V is embedded, the line Lt over k(t) is defined as the line
through the points f(f) and/(r). Its specialisation £q for t = 0 is uniquely de-
fined (consider the morphism of P1 into the Grassmann variety of lines, which
maps t onto the lineZf). Moreover, Lq is tangent to Vat the point x because the
two intersection points /(?) and /(/) coincide at x for t = 0. Because #(0) GZq,
it follows that g(0) G C(x). But according to Lemma 14.3.1, (C(x)\ {*})(£) C
G &(Xf if g(0) = x, then the line Lq either has a threefold tangent point at x,
or it is completely contained in К and again x G
This concludes the proof of Proposition 15.1.
Developing the ideas of this proof a bit further, one can establish in some
other cases that the 3-component of the CML V(k)/R is trivial.
Proposition 15.2. Suppose that there exists a finite extension К D к of к
and a rational map f: P£-> V ® К of finite degree, defined over K, with the
following properties’.
(i) К Dk is a tower of separable quadratic extensions.
(ii)/ is a composite of separable rational maps of degree 2.
Then the CML E = Vfk}jR is an Abelian group of exponent 2.
Proof, Let к denote the maximal separable 2-extension of the field fc Let
E = V(k)jR, V(k)jR and i = i^
: E -> E the canonical
map.
First of all we have that the kernel of i consists of elements of order 2 in
virtue of Corollary 15.1.1.
We now show that Eis trivial, so that E = Ker i^yjf . Indeed, by assump-
tion there exists over к a rational map/: P~ -> V ® к which decomposes as a
composition of rational maps of degree 2. Let UC P77 be a non-empty open
set in the domain of definition of / on which / is unramified. Then, by the def-
inition of the field k, f maps U(k) on all of W(k), where W = f(U). As a conse-
quence, one of the R-equivalence classes is thick in the Zariski topology of
Fr(£), and, as all classes are dense, they all coincide with VT (k).
16. Universal equivalence over local fields. Examples
In this section a local field will be a field of one of the following types:
(i)u The field of real numbers R.
(ii). A finite extension of the field of p-adic numbers Qp.
Let A: be a local field. It carries a topology and is complete and locally com-
pact. In particular, the set of rational points K(fc) of any algebraic variety V
over к is provided with a topology which is induced by the topology on k. We
shall call it the k- topology.
Theorem 16.1. Let V be a cubic hypersurface of dimension > 2 which has
к-points of general type. Then for any admissible equivalence relation on
VT(k) each equivalence class is open and closed in the k-topology.
Proof. It is sufficient to prove that each class is open since it is the comple-
ment of the union of all other classes and hence is closed.
We show first that there exists a set U which is open in the A>topology and
which is completely contained in one equivalence class. The proofs of Theorems
12.11 and 13.1 show that there exists a rational dominating map f: Рл -> V
such that all points in the image f(Pn(kf) are equivalent: in the notation of
Theorem 12.11, the birational fibre space UVG^C(y) is birationally equivalent
to P".
We can assume that n = dim V, by restricting/to a projective subspace of
dimension equal to dim V passing through a point in the domain of definition
of/. As/is of finite degree and separable, there exists a pointy еРл(Л:) such
that / is defined and unramified in y. But then / defines an analytic iso-
morphism of some neighbourhood of у (in the к-topology) with some neigh-
bourhood U of the point x - f(y). This neighbourhood U lies completely in
one class.
We now show that for any other point у E V(k) there is a whole neigh-
bourhood lying in the plass of y. (See Fig. II.4.) The set U is dense in the
Zariski topology. Therefore there exists a non-empty subset if EUwhich is open
in the A;-topology such thatj> is in general position with respect to all points
x E U'. Choose x; then the same considerations show that there exists a neigh-
bourhood U" of the point x о у G which is open in the A;-topology such
that all its points are in general position with respect to x. Moreover, they all
belong to one and the same class by definition.It follows that the set tx(U”)
is an open neighbourhood of the pointy which belongs to the class ofj. This
proves the theorem.
Corollary 16.1 A.Let [fc: Qp] <°° and let Vbe a cubic hypersurface without
singular points. Then the set of classes of points on Vr(k) with respect to any ad-
missible equivalence relation is finite and consists of2a3b elements.
Proof. The set of Appoints Kr(£) ~ V(k) is compact in the A>topology (as a
closed subset in Pn(£)) because by assumption there are no singular points.
Using Theorem 16.1, the finiteness follows from this. Corollary 13.4 yields the
second assertion.
Remark 16.1.2. One can extend the results of Theorem 16.1 and Corollary
16.1.1 to fields of formal power series with a finite residue field. To do this,
we need a more careful analysis of the construction of the rational map
/: Рл -> V to establish its separability. This can in any case easily be done for
characteristic 2, 3.
Corollary 16.1.3. Let к = R. Then universal equivalence and ^-equivalence
coincide on 7(R) and each class is a connected component of U(R). The num-
ber of connected components of K(R) is equal to 1 or 2; the CML E= K(R)/A
is isomorphic to {1} or Z2. respectively.
Proof. It follows immediately from Theorem 16.1 that each class of an ad-
missible equivalence relation is a union of connected components. On the
other hand, an arbitrary rational dominating map/: Pn V completely covers
exactly one connected component of K(R), and the points of different com-
ponents are clearly not R-equivalent. The first assertion follows from this.
Further, if there existed three different connected components of F(R),
then by choosing a points on them and passing a plane through these points,
we would obtain a plane cubic curve with > 3 connected components, which
contradicts Harnack’s theorem.
Example 16.2. Using the results on the calculation of K(R)/R just described,
we shall now show that the number of classes cf Kr(&) can be arbitrarily large
for a fixed cubic surface while к varies over different algebraic number
fields (of growing degree over Q). Let V be such that K(R) = I0 (a split-
ting into two connected components). We shall make use of the fact that the
inclusion Kr(fc) -* Fr(K) sends R-equivalent points into R-equivalent points
for each inclusion of fields к С K.
We now fix some integer n > 0 and suppose that we have constructed an
algebraic number field k, n embeddings sz: к -> R and n points E VY(k)
satisfying
sz(x.)eyocrr(R),
for all i, j = 1,..., n. The points х1? ..., xn E Кг(&) are then pairwise not
R-equivalent to each other because xz, Xy belong to different connected com-
ponents of K(R) under the embeddings sz: Fr(/c) Kr(R).
We now give an explicit construction of the example.
(i) Construction of И Let V be given in affine coordinates by the equation
Г? + 7$ = T2 — 1). This surface is obtained by revolving the curve
Tv = TXTl — 1) round the T? axis. (See Fig. П.5.) Let be the convex (com-
1. *
pact) connected component of Kr(R). The composition law on E~ (Xg, JQ) is
given by the formulae Xj 'о Xj = X& Xy о = Xyf Xq о Xq = Xq .
(ii) Construction ofk, standxt. Let py,... , pn be different primes. For
each i = 1,..., n, choose rational numbers rZp tZ2 £ Q such that
+ >1 >
it is trivial that this can be done. Further we set
= ------------------------------------------—
where
? = Pi , &Г + Ti2 V «ТП + T:2 U2 ) •
Finally, let the embedding : к -+ R be defined by the formulae
s;-(y
s?-(|/) = Vp/, 7*г ;
(arbitrarily extended over Then the points xz- E K(fc) with coordinates
^=0, r2=0;., T’3=7/1+7/2?.
possess the property
s-(x-) EXq , s.(x.)EX^ for
This concludes the construction of the example.
We remark that the equivalence relation Tor all i,Sj(x) is admissible.
The CML constructed from this is Abelian with at least 2n elements. Later
we shall show that the construction of this example can be generalized to ar-
bitrary fields if we use Brauer equivalence.
Example 16.3. We now mention an example of an admissible equivalence
relation which yields'the CML Z3 X Z3. We set = Q2(0), 03- 1 (an unrami-
fied quadratic extension of the 2-adic numbers) and consider the surface V
over к defined by
Tl + T? + T~ + 9 Tl = 0 .
J- Zj «М?
There exists a map ‘reduction modulo 2’:
....................... (IL3)
where P^ = Proj & [7g, Tj, 7^, T3] = F4, the residue field of the ring of
integers of k, and P^ = Proj k$ [Tq, Ту T2, T3]. This map carries a point with
homogeneous coordmates where the E к are integers and rela-
tively prime, into the point with coordinates (/$,... ,73), rj = rz- (mod 2). This
is clearly well defined. Restricting this to F(£), we obtain a map
, (П.4)
where V is given by the equation
+ 9 fj = 0 , 6 = 9 (mod 2) G F4 .
One easily verifies that this surface is non-singular.
We shall prove that the equivalence relation on V(kf
x ~y ффх -y ,
is admissible, and that the corresponding CML is isomorphic to Z3 X Z3. For
this, it suffices to verify the following two assertions:
16.3.1. The collinearity relation induces on V a composition law of an
Abelian symmetric quasigroup, and Т0(К(^0)) = Z3 X Z3.
16.3.2. If the points x, y, z E. V(k} are collinear, then the points
x,y,z E Vik®) are also collinear.
Proof of 16.3.1. For all elements t E F4, we have P = 0 or 1. Therefore
6 E F4 cannot be represented as a sum of three cubes; it follows that all points
of F(fc0) lie on the plane section T3 = 0. This is the non-singular cubic curve
Tq + T’p 4- “ 0; all its points over F4 can be easily enumerated, they are
its nine inflection points. Choosing one of these as the origin, we obtain that
R(&q) consists of the complete set of points of order 3 on a plane elliptic
curve. The desired assertion follows from this.
the product ПД0(а
Proof of 16.3.2. We first of all observe that there are no lines on V defined
over к and also no lines on V defined over k§. In fact, all of the 27 lines on the
surface Я/ ?i = 0 can be obtained as follows: dividing the four terms of the
left-hand side into two pairs and factorizing each sum, e.g.,7’J -ьа^Т^, into
3
1
T + a^tfTy), we represent the equation in the form
LyL^L^ + LyL^L^ = 0. The equations Lj ~ L’j = 0 give 9 lines on V, resulting
from the given division into pairs, and there are in all 3 ways in which to di-
vide them. It follows from this. that in our case all lines are defined over k(fT)
(respectively &о(0з)? but not over the base field. Consequently, the points
x, y, z G V(k) are collinear if and only if there exists a line IC such that
l О V-x + z.
The desired result will be found if we can show that for each line IC
it is possible to define its reduction mod 2, i.e. a line T C p| such that the
following condition is satisfied:
I Ci V ~x + y + z =>T П V =x + y +z . (II .5)
Of course, there exists a general definition of the reduction of a closed sub-
scheme of a projective space, defined over a discretely valued field. We make
do with some special cases of this, treating V, x, I separately.
(i) The construction of I, We consider the set Tkf of all linear forms in
To, Tj, Г2, with coefficients in the ring A of integers in the field k, which
are zero on /. Because A is a principal ideal ring and M has no torsion, M is
free of rank 2 - dim^M 0^ к (becauseM к is the linear space of forms which
are zero on /). Let us denote by M the space of forms of M, reduced mod 2 (we
reduce the coefficients and replace 7} with 7}). We shall show that the system
of equations M ~ 0 gives a line / in P| First of all, dim^ M = 2. In fact, let
nty n^2 GM be a free Л-basis forM If rhy + em2 = 0 for some e = e(mod2)G^0,
then |(m3 + em2) GM, contradicting the choice of/Wp^ and the maxi-
mality of M. Therefore, I is a line since it is the intersection of two different
planes fhy = 0, ~ 0.
(ii) Verification of property (II.5). Let M be the Л-module of linear forms
corresponding to a line Z for which / П V - x + у + z. We consider the ideal
I = (M, Tq + + Tl + 6 T^ in the ring of forms with integer coefficients and
we set В = Л [To, T2, T3]/L
First of all, Proj В & j к is the scheme intersection V A Z, I V, so that
the Hilbert polynomial of В к is equal to 3. Moreover, this is also
true for В к& because this is the scheme intersection V ЛI and 1 V.
Consequently, for i > Iq, B; is a free Л-module of rank 3 (В = ФГ0 Bt, graded
by the degrees of the forms), because 2-torsion in B^ would give superfluous
generators in Bj ®дк§ as compared to В к. It follows from this that
Proj В C P^ is a closed subscheme which is flat over Spec (Л), its fibre over
the general point Spec (k) Spec (Л) is equal to V П I and the fibre over the
closed point Spec (A?o) -> Spec (Л) is equal to V А Г. Thus the underlying space
of V О T coincides with the specialization of the underlying space of V О I.
From this we obtain immediately that V C\ I =x + у + z if x Фу (or, which
is the same, if the points x, y, z are all different). Ifx -y, then alsox~ jT,
and the intersection V П T as a cycle consists of the point x with multiplicity
3 = dim^Bj z0’
This concludes the construction of the example.
I do not know whether the given partition is universal and in which relation
it is to the partition into R-equivalence classes.
17. Bibliographical remarks
This chapter is completely based on the papers Manin [6] and [9]. However,
the definition of admissible equivalence has been changed and universal equiv-
alence has been studied here for the first time. Also, the examples of Section
16 have not been published previously.
CHAPTER III
TWO-DIMENSIONAL BIRATIONAL GEOMETRY
18. The main results
This chapter has an auxiliary character; we have collected in it some in-
formation on the structure of rational and birational maps of surfaces; the
proofs have sometimes been omitted or are incomplete. This information will
be intensively used in Chapters IV and V which contain a ‘subtle’ theory of
cubic surfaces which does not carry over to higher dimensional cases. In the
first section of the present chapter we give a summary of the definitions we
work with and of the results, and some informal comments on them. The
logical presentation in the survey is different from that of the main part of the
chapter.
Let к be some algebraically closed field. We consider only smooth projec-
tive surfaces over k. We denote by Pic V the Picard group of the surface V.
Lemma 18.1.1. Let f: V'-+ Vbe a birational morphism of surfaces and let
f*: Pic V -> Pic Vr, Then Pic V f/f* (Pic K) is a free Abelian group with a fi-
nite number of generators.
Definition 18.1.2. The rank of this group is called the index of the mor-
phism/and it is denoted by i(f).
The map /* : Pic V -+ Pic V' has a trivial kernel, so that the index measures
how much the Picard group grows. It is the simplest numerical invariant of the
morphism /. One easily verifies that z(/o g) - i(f) + i(g).
Lemma 18.1.3. The following statements hold for the index i(fy of the
morphism f:V'^V:
(i) If i(f) = 0, then f is an isomorph ism.
(ii) Ifi(f) = n > 1, then f can be represented as a composition
= к Д к к = 7,
where f is a birational morphism of index one.
It is clear from this that morphisms of index 1 play a fundamental role.
Their geometric structure is described by the following result.
Theorem 18.2. Let f: V' -+ V be a birational morphism of surfaces and let
(i) There exists a closed point x E V such that the restriction
f: V' \ /-1 (x) -> V x is an isomorphism. The point x is called the centre
off Then f~^(x) = D is a curve isomorphic to P1 with self intersection num-
ber -1.
(ii) Let f': V' -> K, f" : V”-*V be two birational morphisms with centre x,
i(f’) ~ i(f") ~ 1 • Then there exists a unique isomorphism g : Vf -> V" such
that f ~ f" о gt
(iii) Let f^ : V’ -> F15/2 : V* ^2 two birational morphisms, i{f\f ~
- *(/2) ~ 1 • Let xt E Vi be the centre off and suppose that ff1 (xj) = =
= D C V*. Then there exists a unique isomorphism g : K2 such that
Wi =/2-
In the situation as described in the theorem, the morphism of index one
f: Vr V is called a monoidal transformation with centre x. The statements
(ii) and (iii) are uniqueness theorems: / is completely determined (up to iso-
morphism) either by V and the centre x E V, or by Vr and the inverse image
of the centre D =/-1(x) C Vr, Therefore the morphism/is also called
‘blowing up the point x’ (from the point of view V) or ‘collapsing the curve
Z)’ (from the point of view F').
Theorem 18.3 (Existence theorem), (i) For every closed point x E V there
exists a monoidal transformation with centre x.
(ii) For each curve D С V' such that D = P1 and (D, D) = — 1, there exists
a monoidal transformation f: F' V with the point f(D) as its centre.
A curve satisfying (ii) is called exceptional.
In fact the theory starts with an explicit construction of a monoidal trans-
formation with the point x as its centre. Thereafter the inverse image f~^(x)
is described. The role of these maps in the general picture becomes only gradu-
ally clear; the formulations cited above are a; survey of the results.
Since we are interested in birational maps (e.g. tx) and not only in morphisms,
we must clear up the place of the latter in the general picture. This is done by
the following theorem:
Theorem 18.4. For every rational map of surfaces : V -> W, there exists
a birational morphism f: V’ -> V and a morphism g : V' -> W such that
This is the simplest form of the theorem on the resolution of singularities
of a rational map; for a more detailed discussion see the first part of Section 19.
Combining Theorems 18.4 and 18.3 gives:
Corollary 18.5. Every birational map can be split up into a product of
monoidal transformations and their inverses^
During the proof of these results, in the main text of the chapter we discuss
another three important themes:
(i) The actions of a monoidal transformation/: V' V on Pic V and
Pic V' on curves of the surfaces V' and V and on their canonical classes (see
the resume in Section 34).
(ii) Galois descent: the transition from the algebraic closure of the base
field к to the base field itself.
(iii) Minimal models.
We shall say a bit more about the latter concept. Let К Э к be an extension
of transcendence degree 2. A model of the field К (over k) is any pair con-
sisting of a (projective smooth) surface V over к and a fixed /^-isomorphism
' k(V) where k(V) is the local ring in the general point of V, that is,
the field of rational functions on V. Two models (Кр<Р1), are ca^e^
equivalent if there exists a (necessarily unique) isomorphism /: -> 72 such
that o/* = ^2 An equivalent definition is: to a model corresponds a set of
local rings in К corresponding to the points of V. Two models are equivalent
if and only if these sets coincide.
Definition 18.6. (i) A surface V is called minimal if every birational mor-
phism f\ V V* is necessarily an isomorphism.
(ii) A model (K, <p) of the fields is called minimal if the surface Vis mini-
mal.
An important fact is that there exists for every surface V' a birational mor-
phism into some minimal surface. The group of automorphisms of the field
К over к has a completely different structure depending on whether there is
or is not a unique minimal model of this field. We investigate first the case
when к is algebraically closed.
18.7.1. The minimal model V is unique. Then it follows easily from the defi-
nition that the group of automorphisms of the field К is isomorphic to the
group of ordinary automorphisms of the surface V, so that birational objects
reduce to biregular ones. Moreover, for surfaces 4of general type’, on which
the sheaf £2^ is ample, the automorphisms of V induce linear automorphisms
of the space 2/°(F, £2y). If n is such that Q/у is very ample, then this yields
a faithful linear representation of the group of automorphisms of V and it is
easy to show that its image is finite.
18.7.2. The minimal model V is not unique. It is known (cf. the book “Alge-
braic Surfaces”) that this is only possible in the case that К has a model of
the form P1 X X, where X is a curve; in other words, К = k(t, u, v), where
f(u, v) = 0 is the equation of the curve X for some polynomial f over k, If
genus (X) > 1, then all automorphisms of К send the subfield -k(u, v)
into itself. On the other hand,/f = K$(t) and the automorphisms of K/K§
are well known: they are the linear fractional transformations t -+(at+b)/(ct+d),
where a, b, c, d EKq. Therefore there is an exact sequence
1 PL(2, £0) Aut (K/k) -> Aut X 1 .
The kernel PL(2, here is infinite dimensional as a group over k', the func-
tions a, b, c, d can have arbitrarily high degree.
Finally, if genus (Jf) = 0, that is, К is the field of rational functions in two
variables, К - k(tQ, z^), the situation becomes even more complicated. One
can conclude from Noether’s theorem that Aut (K/k) is generated by an in-
finite dimensional subgroup PL(2, k(tQ)) and the group of linear maps
(?0> ^)*^(йо^о + й1г1 +j2> ^0^0 + + ^2)-
18.7.3. The geometric picture. Most of the results formulated in Sections
18.7.1 and 18.7.2 are obtained by means of a subtle analysis of the geometry
of exceptional curves. To illustrate the main ideas, we consider two extreme
cases:
(i) . Let 7 be a surface on which there are no rational curves (including
singular ones). Then V is the unique minimal model of its function field. An
example is a surface V which is a product of curves of genus > 1. In fact, let
И7 be another minimal model. There exists a canonical birational map
: V -> W. Let /: V' -> Vf g: 7' -> W be a resolution of its singularities, as in
Theorem 18.4. In virtue of Theorem 18.3,/can be factorised as a product:
of a finite number of monoidal transformations, each of which ‘glues in’ a ratio-
nal curve P1 in the place of a single point. As there are no rational curves on Vy
the glued in curves on V' exhaust all rational curves.
The morphism # collapses some sequence of curves P1. It is evident from
what has been said that ah these must be contained among those produced
by/ But then some combinatorial analysis shows that all rational curves on V'
must collapse, and the image of g is isomorphic to 7.
(ii) . Let V = P1 X P1, and let V* -+ V be a monoidal transformation with
its centre at the point (x, j>) G V. Here there are many rational curves on 7,
and two of these immediately become exceptional on 7', namely, the inverse
images of the fibres x X P1 and P1 X у. [This can be checked as follows: After
a monoidal transformation/: Vf V with its centre at the point x G V of
multiplicity e on the curve D С V; we have
Therefore, if there passes through x a curve P1 of self-intersection number zero,
then it becomes exceptional after blowing up x.] In particular, by collapsing
/”*(х X P1) and/""1^1 X y) we obtain a morphism V' -> P2. The surfaces
P1 X P1 and P2 are non-isomorphic minimal rational surfaces.
Besides these, there is a countable series of non-trivial fibre spaces P^-^P1
with base P1 and fibre P1. The index n refers to the fibre space which has a
section with self-intersection number — n. This is the only curve on Vn
with a negative self-intersection number. Every minimal rational surface
over an algebraically closed field к is isomorphic to either P2 or Vn, where
n - 0 (then V = P1 X P1), or n > 2 (7j is not minimal). The proofs of
these results can be found in the treatise of Nagata [1], in the book “Alge-
braic Surfaces”, Ch.V, and in the paper of Hartshorne [1].
However, the field К ~ &(£$, has considerably more minimal models.
In fact, let (V, </>), where ip : k(V) -> K, be one such model. Then for every
automorphism К %. К we get another model (7, ф</?), so that the infinite
dimensional group Aut(X/£) acts on the left on the set of models. The sub-
group which leaves (У p) invariant is isomorphic to Aut^ V and it is not
difficult to see that it is finite dimensional. Finally, the classes of minimal
surfaces are exactly the orbits under the action of Aut (K/к) on the set of
minimal models of K.
18.7.4. The case of a non-closed field к Let K/k be the field of rational func-
tions in two variables over k. What can be said about the minimal models and
automorphisms of K/kl The situation here is considerably more complicated
and interesting. In Chapters IV and V we shall study the case where К is the
field of rational functions on a cubic surface. Chapter IV investigates in detail
the non-minimal surfaces and Chapter V the minimal ones. Here a whole spec-
trum of possibilities is realized:
(i) The field К can have a unique minimal model. (This is the case if the
cubic surface V is minimal and V(k) is empty.)
(ii) The minimal model can be not unique, although Aut(A/£) acts tran-
sitively on the set of minimal models. (V minimal and V(k) contains points
which do not lie on an exceptional curve of И)
(iii) The minimal model is not unique and Aut (K/k) does not act transi-
tively. (7 non-minimal, for example V is birationally trivial.)
For more details see Section 33 in Chapter V.
19. Monoidal transformations
19.1. Let У, W be algebraic varieties over a field к and let/: VW be some
rational map. Our immediate aim is to study the singularities, the points whem
the map/is ‘not defined’ and, in particular, to learn how to ‘resolve’ them
(in the case that dim V = 2).
A resolution (of singularities') of a map fin the widest sense of the word
shall be every commutative diagram of the form
/4
s / V
v , w
where g and h are morphisms and in addition g is birational.
Adopting this definition, it is not difficult to indicate one general method of
resolution. Indeed, let /be represented by a morphism ip: U Wt where U С V
is a dense open subset, and let C U X W be the graph of We consider the
embedding r^Ct/XPVCFXIV and denote by V' = Гу the closure of in
FX W. It is easy to see that V' does not depend on the choice of 47; it is natu-
ral to call it the graph of the rational map f. The projection morphisms of
FX W on V andiJV induce morphisms g: V* -> F and h : Ff IV; it is clear
that (F' g, h) constitutes a resolution of singularities of the map/. This
method of resolution has two basic defects:
(i) Even if F and IV are non-singular, V' can have singularities.
(ii) The structure of the birational morphism g is not explicit.
Therefore our theory of resolution will be developed by taking some other
ideas into consideration. However, resolution by means of the graph gives us
the basic construction of an important special class of birational morphisms g:
the monoidal transformations with a point as centre.
We start with the following classical example.
Example 19.2 (‘linear projection’). Let An+1 ~ Spec A: [Tq, .., Tn\ be an
affine space, Рл = Proj к[Т$, ..., T'n] the projective space of lines in Aw+1
passing through the origin О = (0,..., 0). The usual ‘projection of an affine
space onto the hyperplane at infinity’
: A«+l {О} P" , *(tQ...................tn} - (Zo:...: f„) , (Ш.1)
represents a rational map /: Aw+1 -> Pw. We shall now resolve the singularity
of this map by means of the graph.
First of all we recall that if : Spec A Spec В is a morphism of affine
varieties, then its graph
C SpecX X Sp6cB ^Spec^ ®
is defined by the ideal (1 ® b ® 1) С A ® B, where <p* : В -> A is the
homomorphism of function rings and b runs through all possible elements
of Я . ..................—..-.-.....-.
Now let
n
p” = и D+(r;) , D+(r;) = Spec к [Т'о/т;,T^] ,
i=0
A"+1=UZ>(7p, D(Tf) = Speck[TQ,... ,Tn, 11Tj].
7-0
Then the graph of <p restricted to X D+(T-) is given by the equation
Г T'
—- —= 0
T2. r;
in Spec £[7^,... ,Tn, 1/Tp> T^/T^ . .., (We have deleted the tensor
product sign for brevity.) It follows from this that in the product
A«+1XP" = Projfc[7’0.,Tn,T'Q,. ..,T'n]
(the ring is graded by total degree in the variables T^}... ,T'n), the graph Гу
of the rational projection is defined by the ideal with the system of genera-
tors (7)7)' - 7)7)'), z, j = 1,..., n. In fact, this ideal defines a closed irreduc-
ible subvariety, coinciding with at all points where the latter graph is de-
fined.
The geometric fibre of the graph over any point except the origin con-
sists of one point, and over О the fibre is isomorphic to P77. Eq. (Ill.1) gives
the geometrical reason for this: if the point (r0,... , tn) approaches the ori-
gin, while remaining on one and the same line through the origin (or, more
generally, on'a curve with tangent), then the point ^(r0,. .. tn) approaches
the point of the ‘glued in’ fibre corresponding to the direction of this line.
In other words, to resolve the singularity of f we replace the point where f is
not defined by the whole projective space of tangent directions at that point.
It is not difficult to verify that Гу is a non-singular variety.
The significance of this example for the general theory is illustrated by the
following circumstance.
Let V be an affine variety, x G V a fc-point on it, and a closed em-
bedding such that x is mapped onto the origin. Then the composed rational
map ‘linear projection of V from the point x’, Ал+1 -> Pn, is defined. We
can again try to resolve the singularitiesjof-this map with the graph method,
and then we obtain a diagram ( V\ g, h). It turns out that this diagram does
not depend on the choice of the embedding 7е* Ал+1 (up to a uniquely de-
fined isomorphism) but is defined by the point x G V alone. The morphism
g : V' V which we obtain by means of this construction is called the
monoidal transformation of V with centre at the point x.
To prove the invariance, we first give a new construction of monoidal trans-
formations, which will be invariant by definition, in a significantly more gen-
eral context. Thereafter we show that this coincides in our particular case with
the graph of a linear projection.
Lemma and Definition 19.3. (i) The local version'. Let A be a ring. IC A
an ideal, and
R= Ф/К-{2к>од Г1аке/к}сл(Л
к=0
a graded ring (the variable T defines the grading). Then the embedding A^R
defines a morphism
Г = Proj A И = 8ресЛ ,
which is called the monoidal transformation of the scheme V with its centre
in the closed subscheme Spec (A/I).
(ii) The global version'. Let V be a scheme, I C Oy a sheaf of ideals and
R = IK a sheaf of graded Oy-algebras on V. Then there exists a uniquely
defined morphism
g: K'=Proj
which locally on V gives the morphism constructed under (i). It is called the
monoidal transformation of V with centre in the closed subscheme W defined
by the sheaf of ideals I.
Let V be a variety over a field k. Then Vr is also a variety and g is a biratio-
nal morphism which induces an isomorphism V* g’”1(^) 5 F\ W.
Let W be a к-point of V (with its structure as a closed subscheme), V an
affine variety. Then Vf (together with g) is isomorphic to the graph of the
linear projection of V from the point x.
Proof. To carry out the global construction of monoidal transformations,
one must verify, in terms of (i), that for any multiplicative system 5 C A and
any ideal /$= Im (4$ 0^7) G4$ there is a:natural-isomorphism”™.
OO - J oo
Ф /к )®ллД.
к=0 ”k=0 f h
This guarantees the possibility of gluing together the affine pieces. The de-
tails are left to the reader.
If the ring A has no divisors of zero, then the same is true for R. If A is
Noetherian and I = (Д,... ,fn\ then R - Ф~-о TK is generated by the
elements T,... ,fnT over A In particular, Fr= Proj7? is a variety if
F= Spec (4) is a variety.
Every point x G V \ W has an affine open neighbourhood U= Spec A on
which Z Ij; = 0The restriction g ]g-i (/7): g^t/) Uto this neighbour-
hood is isomorphic to the canonical isomorphism Proj A [ T] Spec A. The
monoidal transformation is therefore an isomorphism outside W.
Finally, we prove the last assertion; it is local on V. Let V = Spec4; choose
an embedding Spec fc[T0,..., Tn] for whichx goes over in the origin.
This is equivalent to choosing a system of generators (f^,... , fn) in the \
fc-algebra A such that (^,..., fn) - IQ A is the ideal of the point x Arguing
as in Example 19.2, we find that the graph of the linear projection of V for
this inclusion is embedded as a closed subscheme in
Proj л [ r0,..., ти]/(... ...) -
There exists a natural homogeneous epimorphism of A -algebras
oo
Л[Г0....®I*TK : class of 7} » f.T.
к =0 •
To conclude the proof of the lemma one must only verify that this induces an
isomorphism of Proj (®^_q ZK TK) with the graph.
In fact, this epimorphism of rings induces a; closed embedding of V' into
the graph of the linear projection of which the image is everywhere dense. Con-
sequently, the image of V' must be defined by a nilpotent sheaf of ideals, which
can only be the zero sheaf because the graph is a variety. This proves the lemma.
We shall now study the local structure of a monoidal transformation of a
variety V with its centre at a non-singular closed point. The geometric meaning
of the theorem following below is that, as in thexase of the linear projection
in P”, at the place of x a projective space (of tangent directions at x) is glued
in, and its embedding in V has the standard conormal sheaf.
Theorem 19.4. Let V be an n-dimensional variety defined over a field k,
x EV a regular closed point with residue field k(x), and f: V* -+V a monoidal
transformation with centrex. Then the following statements hold'.
(i) Wr = /-1(х) - V‘ X yk(x) is isomorphic to as a к(x)-scheme and
it is locally defined by a single equation in Vf(i.e., W' is a Cartier divisor).
(ii) Let Z' C Oy* be the scheaf of ideals defined by W' in V'. Then the
conormal sheaf invertible and isomorphic to the standard sheaf
Vi(1)*
(iii) All points of W' are regular on V'\ in particular, if V is regular, then
V' is also regular,
(iv) If V is a projective variety, then V1 is also projective.
Proof. All assertions except the last one can be verified by replacing V by
a sufficiently small affine neighbourhood of x.
(i) . Since the local ring 0x of the point x is regular, we can choose n ele-
ments ty,... , tn Emx which generate the maximal ideal mx C 0x. There-
after we extend all f to a section of the sheaf Op in some affine neighbour-
hood ofx and we shrink this neighbourhood until (/p ... , tn) exactly gen-
erate in it the sheaf of ideals of the point x. Thus we get the following
situation: V = Spec?l, I = (f,..., tn)CA is the ideal of the point x,
dim/f V - n.
We have
Vr = Proj / Ф Zxj ,
' к =0 /
and
W' = V' ®Ak(x) s Proj( © (/* Q^/nUproj ( © Zk//k+M .
\ K =0 f ' к =0
[To prove the last isomorphism, we take the tensor product over A of the exact
sequence 0-> I-+А ~>A/I 0 with IK; this gives I IK IK '-» IK ®aAJI 0,
from which I* §§aA/I.} Therefore, to prove the isomorphism
Wf = Р£(^) is sufficient to verify that ®“=sq/k//k+1 — (the sym-
metric algebra). [Indeed, it follows from the fact that x is regular that
dim^^///2) = n, because Z/Z2 is the Zariski tangent space.]
There exists a natural epimorphism of rings
•’ Sk(xtIl1^ Ф /K//K+1 ’
k=0
(II1.2)
which for elements a^,..., aK G/ and a. = zzz (mod T2) is defined by the for-
mula
<p(a, ... a ) = a.... a (mod/*+1) .
1 /V 1 Л
One immediately verifies that is well defined. Further, both rings have a natu-
ral grading, so that to prove that is an isomorphism it is sufficient to verify
that the homogeneous components of the same degree are of equal dimension:
dimjtws-w2)= ("+* J) = d«WK//K+1 •
The last equality follows from the fact that the completion of 0x is iso-
morphic to the ring of formal power series in , tn> because 0x is regu-
lar, and Z*/Zx+1 es mx/mx+1. Hence W' = P/фф Further,
Here
Z>j7zT) = Spec
Ф
к-0
- Spec?!
ЩТ)
(the adjunction of t^/ti to A takes place within the fraction field of the ring?!).
The ideal defining W' within is generated by the elements zy/l,
j = 1,..., n, that is, by the elements Zz-/1 and Zz/1, tj/ti9 / Ф z. But the ele-
ments zy/Zj are regular in/)+(zz-T), therefore zy/1 = 0 gives the local equation
of W’ within K'. This proves assertion (i).
(ii) . We have already established that W' is locally given by one equation,
which is not a zero divisor. From this it follows already that the sheaf 171*21 IV'
is locally free. Identifying Wf with by means of the isomorphism (Ш.2),
we obtain that I/T'2 glued together from the structure sheafs on
IV1 П D+(tjT) = Proj k[tjtb . . ., by means of the transition func-
tions Zz-/Zy, but this cocycle describes the sheaf 0(1), as is well known.
(iii) . To prove the regularity, it suffices to check this separately on the
open setsP+(Zz-T) = SpecA [Zj/Zz,., Zw/zJ. An arbitrary point of W' lies in
the closed subset Spec (A \t-Jtj,..., Zn/zJ /(zz-/1)) = (a Part °f the pro-
jective space P£j“* on which one coordinate function is not zero). Since P^~*
is regular and Zz/1 is not a zero divisor in A , Zn/Zz], the desired result
follows from this.
(iv) . We restrict ourselves to the case that k(x) = k, that is, the point x has
degree 1. Let КС P^ be some projective embedding. In Section 19.3, we es-
tablished that K' is isomorphic to the graph of the linear projection of К into
P^-1 from the point x Therefore V' С V X P^-l С X P^~l, and the
last product, as is well known, is projective. This proves the theorem.
Remark 19.5. It is clear from 19.3 that monoidal transformations are
compatible with the extension of the base field in the following sense of
the word. Let К be a variety over k, W C Ka closed subscheme over k, and
f: К' К the monoidal transformation with centre W. Then for every exten-
sion of the field of constants К D к the morphism / 0K : V'®k~>V0K
is the monoidal transformation of V 0 К with centre W ® K.
In particular, let W ~x be a closed point and let the extension k(x) be sep-
arable. If К is a Galois extension containing k(x\ then IV ® К С К ® К is the
disjoint union of [fc(x) : fc] closed points (as a subscheme), and the monoidal
transformation with centre W 0 К is a succession of monoidal transforma-
tions with these points as centres, composed in any order. (We suggest that
the reader specifies this statement.)
Conversely, let К D к be a separable closure of the field k, and let there be
given on V ®К a closed subscheme W' which is a disjoint union of a finite
number of closed points which are conjugate over к (in the sense of the
action of the Galois group on V' 0 k К through the second factor). Then
IV' - W0fcK, where IV С К is a closed point, and the monoidal transforma-
tion with centre W' is obtained by extending the field of constants to К out
of the monoidal transformation with centre Ж
These considerations permit us in particular to pass from the case of a per-
feet field к to the case of an algebraically closed field, which gives the possi-
bility of an account in terms of geometric points.
20. Monoidal transformations and divisors
In this section we shall study the effect of monoidal transformations on
divisors and invertible sheaves over an algebraic surface V defined over a field k.
We recall only the basic facts from divisor theory; for details we refer to
Mumford’s book [2], lectures 9, 10 and 12.
20.1. Effective divisors. An effective (Cartier} divisor on К is a closed sub-
scheme D for which the defining sheaf of ideals is locally smooth. (Since V is
a variety, in particular an integral scheme, this requirement is equivalent to
the invertibility of the sheaf of ideals.) Then the support SuppD is a (gener-
ally speaking irreducible) curve or empty. Divisors can be added by multi-
plying their ideals.
Suppose that we have two divisors Dp D2 C 7 such that the intersection of
their supports is zero-dimensional. Then the intersection as a scheme
(it is defined by the sheaf of ideals I t + 7 2, where IK defines DK) is affine
and zero-dimensional over k: “ Spec A, where A is a finite-dimen-
sional ^-algebra. The number
(Dp Z>2) = dim^ A
is called the intersection number of the divisors andD2. The intersection
number is invariant with respect to extensions of the field of constants К Э к:
(D. K} = dim„(A 7C) = dim.X = (DnD0).
Ж XY /V Ж
Because Я = (where Ax is the local ring of Spec A in the point
x G Spec Л), we have
(D.,D2)= S dimkAx= Z) (7)p D^x .
xeSuppZ)jGD2 xeSupp/^nZ^
The ring Ax can be calculated in the following manner: Let (resp. r2) be the
local equation of the divisor Dy (resp. D2) at point x, that is, I у x~$xh>
22Л = Qxh-111611
Ax ~ ^x^l’ (2^ ’
The dimension of this fc-algebra is the local intersection number of Dy and D2.
It is equal to zero if one of the elements ty, /2 is invertible in 0x, i.e., if the
divisor Dy orZ)2 does not pass through x. It is equal to one if k(x) = к and
(tp Г2) = mx (the maximal ideal in 0x), i.e., if ty (mod mx) and t2 (mod тх)
are linearly independent over k(x). This means thatDj and D2 intersect each
other transversally at the point x, i.e., they have different tangents in x. In
general,
(Dp D^)x ~ [&(x): k] dim^) 0xl{ty, Z2) .
The first factor here reflects the circumstance that the point x decomposes
into [fc(x) : k] geometric points over the closure of the field A; (if k{x}!k is
separable); the second factor describes the intersection number of the divisors
Dy and Z>2 in these geometric points.
20.2. Let/: V1 -> К be any morphism of surfaces, D С V a Cartier divisor.
If/(F') t Supp D, then the inverse image f*(D) is defined. It is a Cartier
divisor on 7/if D is defined by the sheaf of ideals I C Oy, then/*(/)) is de-
fined by the sheaf of ideals/*(1) C Oy'. We shall now investigate this inverse
image operation in the case where/is a monoidal transformation at a closed
point. The result can be more conveniently formulated if we make use of the
definition of a Weil divisor. An effective Weil divisor on V is a linear combina-
tion of irreducible curves (that is, of one-dimensional closed subsets) on V
with non-negative integer coefficients (as an element of the free Abelian group
generated by all such curves).
If V is a non-singular surface, then the semigroups of Cartier and Weil divi-
sors are isomorphic. In fact, in this case any irreducible curve on V is locally
given by one equation (as a reduced subscheme) which permits us to construct
a Cartier divisor from a Weil divisor. The construction of the inverse map
rests mainly on the fact that the local rings of V are regular, and that therefore
the factorization of polynomials in them is unique (cf. Mumford [2], lecture 9).
Locally, therefore, every Cartier divisor can be uniquely represented in the
form of a linear combination of Weil divisors ,given by prime elements.
Definition 20.3, Let V be a non-singular surface over k, x G V a closed
point, D С V an effective divisor, and V' V the monoidal transformation
with centrex On V' there exists a unique (Weil) divisor (the
Dj are irreducible curves) such that:
(i)Saz/(^-)-^;
(ii) the coefficient of/"1^) т/~Ц/>) is equal to zero.
The divisor jf”1 (D) is called the proper inverse image of D.
In other words, the map is a homomorphism of semigroups which on
the generators is defined in such a way that for every irreducible curve D С V
the curve Z"1 (D) is the closure of the inverse image ofZ) I V'
It is clear that we have =f*(D) for every divisor D С V for which
x Ф SuppD. However, if x G SuppD, then/*(£>) = + some multiple
of/”4^). The divisor(x)is a projective line on K'; we shall presently des-
cribe the difference — f~\D) and the behaviour of/~x(D) in the neigh-
bourhood of/“4-*)* To be able to see things better geometrically, we restrict
ourselves to the case of an algebraically closed field k. The reader can without
difficulty supply the changes necessary in the general case, but for our objec-
tives this result will be sufficient in view of the compatibility of the basic con-
structions with extensions of the field of constants.
Let t G 0x be a local equation of D in a neighbourhood of x We designate
by e = e(x, D) the number such that
t G me \ ,
x x
The number e is the multiplicity of the point x onD.
Further let T = t (mod m^+1) G = Se(jnx/m^) (cf. the proof of
Theorem 19.4 (i)). In other words, T is some form of degree e in two variables
(a basis of mx/m^ over &), the ‘leading form’ of the local equation of the di-
visor D. We have therefore (using that к is closed)
i~l
where the Emx/mx are different elements of the Zariski tangent space,
> 0 and Sf=1 ez- = e. Geometrically this means that the divisor D in a neigh-
bourhood of x ‘up to a higher order of smallness’ behaves itself as the union
of an -fold curve f = 0, an C2“fold curve /2 “ 0, and so on. Moreover, x
is a simple (of multiplicity one) point on each of the curves Zz- - 0 (the equa-
tions of these curves are only defined mod m*f We shall therefore describe
this situation by saying that the divisor!) has r different tangents at x (with
multiplicities е±, ..., er, respectively).
Proposition 20.4, Let xEVbea regular closed point on a surface defined
over an algebraically closed field k, let ft V’ -+V be a monoidal transformation
with centre x, DC Van effective Cartier divisor and x C Supp D; Wf =
~ f~\x) ~ Z/ie inverse image of the point x, is a Cartier divisor on Vf.
The following statements hold'.
(i) f*(D) = f^iP) + e W', where e is the multiplicity of x on D.
(ii) Let D have r different tangents at x of multiplicities e^..., er. Then
the intersection f"^{D) Л W’ consists ofr closed points , xfr which
naturally reflect the tangents ofD at x, and
,=e
xi
(Informally speaking, the monoidal transformation ‘splits up’ the branches
of D passing through x in different directions.)
Proof. Since/induces an isomorphism Vf \ Wf V and as W' is
irreducible, it is clear that f*(D) — f^(D) is some multiple of We need
only show that this multiplicity is equal to e. Replacing К by a sufficiently
small affine neighbourhood of x, and by its inverse image, we can assume
that the following conditions are fulfilled:
(a) V = Spec A; the ideal of the point x in A is generated by two elements
ty t2 CA.
(b) The divisor D on V is given by one equation t = 0, t C A; moreover,
?2)^(Г1, f2y+l.
Then V' = Spec (А [tj/Г2]) U Spec (A [t2/q ]) and W' is given on these
two open sets by the equations t2 = 0 and t1 ~ 0, respectively. It follows im-
mediately from condition (b) that
t CA [t./t^] t% ,
* V 2J 2 7
teA .
Since z/1 = 0 is the equation of f*(D) within V\ it follows from this that W' turns
up in f*(D) with multiplicity at least e. If this multiplicity were greater, then we
would have t GA [Zj/z^J ^2+1- shall show that it would"follow from this
that t G (Zp Z2)e+1, which contradicts our assumptions. In fact, A [Zj/Z2] ~
— A [T]/(Z2T- Zj), hence A [Z1/Z2]/(Z1/1, z2/l) = &[Т], where к ~ k(x) and
T is the class of Zj/Z2 modulo the ideal (Z^/l, Z2/l). If te^ divides t, then
Z/Z* G^/l^/OCA^/zJ ,
but denoting by z = /(Zp z2) = z(mod mex+1) the leading form of t as a poly-
nomial in 7p Z2 (Zf = Z;(mod mx)), we have
/(T, n-a/zpCmod^/Lz^i))
in A (ZX/Z2J /(Zj/1, Z2/l). The desired result follows because Z Ф 0.
We shall now calculate the intersection Z"1 (D) П W'9let us say within
Spec A[Zj/Z2]. The divisor f~\D) is given by the equation z/z2 = 0 and the
divisor W by the equation Z2/l = 0; therefore
П W' ~ SpecA[Z1/Z2]/(Z2/l, Z/Z2)
= SpecA[z1/z2]/(z1/l,Z2/l,z/zp .
Taking first the quotient by (Z^/l, Z2/l) and then by Z/Z2 mod (Z^/l, Z2/l), we
obtain, in the previous notation,
/“1(Z))nif/ = SpecA;[n/.f(T,l) .
The various zeros of the polynomial f(T, 1) give the various points of the in-
tersection, and the multiplicity of each of them coincides with the multiplicity
of the corresponding zero. This completes the proof of the proposition.
20.5 . The semigroup of effective divisors on V is included in the group
Div V, the elements of which are called divisors. Let/: Vr К be some bira-
tional morphism defined over k. Then the map /* can be extended to a group
homomorphism /* : Div V -> Div V', and its image has a canonically defined
complementary direct summand in Div V*. Indeed, we define a homomorphism
f : Div 7'^ Div V,
in the following way: LetD С И'Ьеап irreducible curve; then f*(D) is either
the curve/(£>) (the set-theoretical image) or zero if f(D) is a closed point. It
is obvious that/* о f* is the identity map, so that
Div 7'=/* (Div 7) ©Ker/* .
In the case where/is a monoidal transformation with its centre at a closed
point, Ker/* = Z FV', where Wf is the inverse image of that point. In the gener-
al case, Ker/* is generated by the collapsed curves (with respect to/). The
maps/*(resp. /*) transform Div into a contravariant (resp. covariant) functor
on the category of surfaces with birational morphisms.
20.6 . The Picard group. Let D be a divisor on V, U С V an open set, t a
local equation for D on U. Those open sets on which DI is given by one equa-
tion constitute a basis for the open sets, and the presheaf
и^г'ци, ov)
which is naturally defined over this basis is a sheaf. It extends uniquely to an
invertible sheaf on 7, which is denoted by 0 y(D). The map D ь* class of
0 y(£>) defines an epimorphism of groups Div V Pic V; the kernel of this
map consists of the ‘principal’ divisors which can be given by one equation
globally.
Suppose that D is an effective divisor. Then 0 y(—D) is a coherent sheaf
of ideals on 7, and there is a canonical exact sequence
0 0y(— D) Oy 0D 0 ,
where 0$ is the structure sheaf of D as a closed subscheme of V, In the case
where V is a projective smooth surface, this gives the possibility to point out
an important cohomological description of the intersection number.
Let L be an invertible sheaf on 7; we set
X(L) = dim. H°(V,L) - dim. Hl(V,L) + dim. H2(V,L).
/V /v rv
Then we have the following:
Proposition 20.7. Let Dp D2 C V be two effective divisors on a smooth pro-
jective surface V and let their intersection be ^-dimensional. Then their inter-
section number is equal to
(Px, d2) = x(0v) - x(0K(-^)) -- x(0v(-d2)) + х(0г(~л1 -P2)) •
For a proof see Mumford [2]? lecture 12. It is based on the fact that
(D D2) = dimtf°(F, 0„ ® 0z>J = x(0n ® 0D ),
and on the existence of the locally free resolvent for Oty which was described
above.
The role of this result is that it permits us to give a canonical definition of
the intersection number as a scalar product on the whole Picard group, and
also on all divisors, even if their intersection is not 0-dimensional.
Lemma and Definition 20.8. Let L^, L2 be two invertible sheaves over a
smooth projective surface К We set
(ZPZ2) = x(0y) - x^r1) - X^J1)+ x^r1 ® Ч1) •
This 'intersection number is a bilinear symmetric scalar product, Pic KX Pic
which coincides with the geometric intersection number of divisors in those
cases where the latter is defined:
(0r(D1),0r(D2)) = (DpD2),
and it can serve as a definition of (Dp D2) in the general case.
As regards the proof, we again refer the reader to the lectures of Mumford.
Now let f: Yr -> V be a monoidal transrormation with its centre at the
closed point x, and let W' be the inverse image of this point.
Lemma 20.9. f induces an embedding of Picard groups f* : Pic V Pic V1
(corresponding to the homomorphism f* of the divisor groups) which preserves
the intersection numbers’.
(Г^рГА2) = (Хр£2),
for all Ly Eric E Moreover;--------- ~ ~ ..
(W'f Wf) = - d, d = [Цх) : к] , (/* Д PF7) = 0 ,
for all L GPic V (we write W ' instead of Oy (JV')).
Proof. The sheaf f*(L) is invertible on Vf for every invertible sheaf L on И;
in particular, if L = Oy (D), where D is an effective divisor, it immediately
follows from the definitions that/*( 0 p(/))) is canonically isomorphic to
Oy(f*(D)\ Therefore the maps/" on the groups Div and Pic correspond.
The following general observations are useful for the calculation of (W’> И7').
Let L be an invertible sheaf and D a curve on V’. Then
(L, 0v> (£>)) = deg(L|p) = x(0D) - х(Ь~г |p)
(see Mumford [2].lecture 12). We now take/) = W' and A = OyIt fol-
lows from Theorem 19.4 that the curve W' is isomorphic to , so that
xCOjt/') = d. (Note that x is calculated by taking dimensions over k, and not
over fc(x)!) Moreover, the sheaf of ideals T defining W' С V' is Oy(—W’f>
therefore, L-1 |и" = T\wf — 17Г2 — Opi(l) (Theorem 19.4(h)). Consequent-
ly,
X(L-1lr) = x(Opl (l)) = dim*tf°(pl 0(1))-dim Я1^)’
from which, finally,
(0K'(^'), 0V’(W')) = -d.
Now let D С V be an effective divisor and e the multiplicity of the point
x on D, According to Proposition 20.4,
/* (/))=/-!(/))
and
(rl(P),W') = ed.
[This has only been proved for d = 1; the general case immediately follows
from this by extending the field of constants if k(x)/k is separable; or it can
be verified by changing slightly the arguments of Proposition 20.4.] We find
from this
W’) = (P1^), W') + e(W', W') = ed - ed = 0 .
Because the classes of the sheaves f*(D) generate f*(Pic K), we find that this
group is orthogonal to Finally, let Dy, be effective divisors on V such
that their intersection is 0-dimensional and does not contain x. Then
(Z>1,D2) = (/-1(£>1),/-1(D2)),
because H £>2 *s contained in the open set V \ {x} and the restriction of
f to this set is an isomorphism. Further
CT1 ), f -1 (Z>2)) = CH^) - elw ’ f* (D2> - e2 W')
= (ГФ1),Г(^2))>
because at least one of the two multiplicities elf of the point x on D±,
is equal to zero, and W’ is orthogonal to/*(Pic F). But the sheaves
(?pr(jD), where/) does not pass throughx, generate Pic F. (The orthogonality
of/*(Pic V) and IP' also follows immediately from this remark.) This con-
cludes the proof of the lemma.
Corollary 20.9.1. Under the conditions of Lemma 20.9, we have
Pic F'=/* (Pic F) ® Zw,
where w is the class of Oyt(Wl) and the subgroups f* (Pic F) and Zw deter-
mine one another uniquely as orthogonal complements with respect to the
intersection number.
Proof. As was remarked m 20.5,
Div F'=/* (Div F) ®ZW'.
The kernel of the canonical homomorphism Div K'-> Pic V1 is completely con-
tained in/* (Div F), because if f*(D) + aW’ is a principal divisor, then
0 = (/*(D) + aPF', W') = -a.
Consequently, Pic Vf = /*(Pic V) © Zw. The second assertion also follows
immediately from the lemma.
The last result of this section describes the behaviour of the canonical
class go у GPic V under monoidal transformations.
Proposition 20.10. Under the conditions of Lemma 20,9, we have
where w is the class of the sheaf in Pic V’.
Proof. It will be convenient to use the following constructive definition
of the class co. Let П G Q2 (k(v)/k) be some exterior form of degree 2 in the
field of rational functions on V. For every closed point x G V, we define the
divisor^ of the form £2 in a neighbourhood of x by the following condition:
if £2 = tdzy Л dz2, where z у z% £ tnx are local parameters, then t is a local
equation of К at x. That this is well defined follows without difficulty from
the invertibility of the Jacobian under a change of the system of local param-
eters. Under these conditions we have
coy - class 0p(K).
Therefore
f*(uy)= class 0r(/*(/O).
On the other hand, gov> - class 0yf(Kwhere Kf is the divisor of the form
/*(£2), the canonical image of £2 in the field k(vr) of rational functions over V\
which is identified with к (u) by means of/. The form £2 can be chosen such
that x is not contained in the support of K, Then it is clear from the defini-
tion that f*(K) =f~l (K) and that K'~ f*(K) is some multiple of PF". To cal-
culate this multiplicity, it suffices to consider the situation locally in a neigh-
bourhood of a closed pointy G W',
Let Zp z2 be local parameters inx, £2 = tdz^ A Jz2, t invertible at x. It is
clear from the proofs of Theorem 19.4 and Proposition 20.4 that (zj/z2, z2)
forms a local system of parameters in a neighbourhood of a typical closed
point on W'. Therefore we have
/*($2) = tz2 dizjzf) A dz2 ,
and because t is invertible at x, tZ2 is a local equation of W' on V' together
withz2. This proves the proposition.
21. The main theorems on birational maps
In this section we shall formulate, guided by the ideas and results introduced
above, the main technical statements, which afterwards shall be used in the
birational theory of cubic surfaces.
Theorem 21.1 (Resolution of singularities of a map). Let V be a smooth
projective surface over a field k,W a projective variety and f:V~*W some
birational map. Then there exists a resolution off
(cf Section 19.1), in which g decomposes as an iteration of monoidal trans-
formations with their centres at closed points which lie over points where f
is not defined.
Proof (outline). First of all, any rational map f: V W of projective
varieties is defined in all normal points of codimension 1. For the proof of
the general case one first reduces to the case W = P1; we omit this reduction.
If W - pl, then /is represented by a rational function y? on V; at any normal
point x of codimension 1 on K, the function is either defined or it has a pole,
because is a discretely valued ring, and poles correspond to the point at in-
finity on Р1. Therefore, points where/is not defined (in which the function
leads to 0/0) do not exist. In particular, if V is a smooth projective curve, then
every rational map of it is a morphism. If И is a surface, then the points where
У is not defined are isolated. To resolve f; it suffices to consider them separa-
tely; moreover, we can again assume that W = P1, so that we need to resolve
the points where a rational function is not defined; to do that, it suffices to
work locally. In order to illustrate the principles of the proof, we consider
some of the simplest cases:
Let be a rational function on V and x G V a point where is not defined.
For simplicity, we shall assume that к is algebraically closed (or k(x) = k).
Further, let the divisor of in a neighbourhood of x be equal to Dq — D^,
where and are effective divisors which do not have a common compo-
nent. The character of the singularity of x is determined by the complexity of
the behaviour of Z>0, DM at the point x. Let us consider the behaviour of
and its divisor under a monoidal transformation f: V' -+V with centrex
under various partial assumptions with respect to Dq and Dx.
Case 1. The multiplicity of x on Dq, Dx is equal to i,andDq, Dx are not
tangent to each other at x. Then the divisor of on V’ is equal to
/* (Do) - f*(D„) = (/“' (Do) + W') - (Г1 (D,)+ И/)=/-1(П0) - r\Dx )
in a neighbourhood of W' = /-1(x); and/-1 (Po),/-1 (£>«,) no longer inter-
sect in a neighbourhood of И7' (cf. Proposition 20.4). In this simplest case,
therefore, the point of non-definition is removed by one monoidal transfor-
mation.
Case 2. The multiplicity of x on Dq andis equal to 1, but Dq andDx
are tangent to each other. In this case, after a monoidal transformation with
centre x, the curves/-1(D0) and/-1 (£<*,) will intersect each other in a neigh-
bourhood of W' at precisely one point, as before, but their order of tangency
does not increase, and after a finite number of repeated monoidal transforma-
tions it diminishes.
Case 3. Dq has one tangent of multiplicity at x and Dx has one tangent
of multiplicity at x, and these tangents do not coincide. After a monoidal
transformation, the divisor of will have the form
locally in a neighbourhood of W'.Tor Cq = e^, the point of non-definition
vanishes; for Cq > ем it emerges at a point of the intersection of /“^(D*,)
with W', and for at a point of the intersection of with Wf,
but the multiplicity of one of the components of W' will be less than the maxi-
mal multiplicity at x. — - ~ ~
These examples give a sufficiently clear picture of the character of the
‘simplification’ of points of non-definition and of the invariants which must
be watched. For more details we refer the reader to the lectures of Safarevifc
[2] (surfaces over an algebraically closed field) or [1] (smooth 2-dimensional
schemes).
Example 21.2. The structure of the map tx. Let V С P3 be a smooth
cubic surface, x G V(k) a Appoint of general type. The map tx: F V was
described algebraically in Proposition 12.13. In particular, it induces an automor-
phism on the complement of C(x), the intersection of the tangent plane at x
with V. The geometric map tx looks as follows:
Let Vf £. Fbe the monoidal transformation with centre x. Then there ex-
ists on Vr an automorphism of order 2, tx: Vf -* V\ such that tx о f - f о tx.
Moreover f induces an isomorphism
and on this open set t’x acts precisely as tx on K^C(x). In addition, t'x inter-
changes Wf and/-i(C(x)). These two curves intersect at two different geo-
metric points (as shown in Fig. Ill.1) if C(x) has distinct tangents at x. If x is
Fig. III. 1.
a cusp for C(x), then W' andf ”ЦС(х)) are tangent to each other at exactly
one point.
A different method of lookingat the map , which is often useful, is the
following: Let P2 С P3 be a plane which does not pass through x. The projec-
tion from x of P3 on P2 is a rational map which is resolved by the monoidal
transformation U* P3 with centrex; it defines a commutative diagram
The composed map V* P2 is a morphism of degree 2, and tx is the automor-
phism: ‘interchange the sheets of this two-sheeted projection’. As an exercise,
we suggest that the reader should translate these assertions into algebraic lan-
guage and verify them.
Control questions'. How can one resolve the singularities of the map tx in
the case when x is not a point of general type on the surface? Can the map
tx be a morphism? (Answer: Yes, if C(x) consists of three lines intersecting
at the point x.)
Example 21.3. Stereographic projection. Let V С P3 be a smooth quadric.
It is well known that it is isomorphic to P1 X P1 (we here take the base field
to be algebraically closed). The intersection C(x) of the surface V with its
tangent hyperplane at an arbitrary point x G Г consists of two lines intersecting
each other at this point; these are the fibres with respect to the two projections
of V on P1. Let P2 С P3 be a plane which does not pass through x. We con-
sider the rational map
which is a morphism outside C(x) and which is defined by: the points x, у G V
and r(x) G P2 are on one line. Its resolution looks like
where / is the monoidal transformation with centre x, and g is the monoidal
transformation with as its centre the two points of the intersection of P2 with
the components of C(x). In other words,/blows upx, and g collapses the
Z”1 inverse image of C(x). This can be easily proved by a local analysis. We
remark that the components of C(x) are isomorphic to P1; their self-intersec-
tion numbers on V are zero; after being lifted to V\ they become — 1, which
easily follows from Lemma 20.9.
Theorem 21.4 (Structure theorem for birational morphisms). Let /: V W
be a birational morphism of smooth projective surfaces over a field k. Then
f is an iteration of monoidal transformations with their centres in closed
points. In other words, there exists a sequence of surfaces and morphisms
V = v0 \ f-X v2... V = w
such that/•: -> Kz- is a monoidal transformation with its centre at a closed
point xz- G Vi and such that f = fr ° fr_y о ... оД.
We omit the proof (cf. Safarevi^ [1] and [2]). However, its main feature
is needed for further applications, and therefore we formulate it separately.
Lemma 21.4.1. Let /: V^W be as in Theorem 21.4 and let the rational map
f~l be not defined at the closed point x G W. Then f decomposes into a prod-
uct V W' W, where h is the monoidal transformation with centre x,
and f is some morphism.
Informally speaking, a ‘not being defined’ of’the type of a ‘monoidal
transformation’ is the most economical kind of ‘not being defined’ of a ratio-
nal map.
From Theorem 21.4, we immediately obtain the following useful corol-
lary:
Corollary 21 A.l.Let /: be a birational map of smooth projective
surfaces. Then in every resolution
Г
g / h
v —L „ w
off the morphisms g and h are iterations of monoidal transformations with
their centres at closed points, provided V' is smooth.
Examples 21.2 and 21.3 and the answers to the control questions can
serve as illustrations of this general principle.
Theorem 19.4 describes the structure of exceptional divisors, the inverse
images of a point x under monoidal transformations with centre x. One of the
useful results is a partial converse to this theorem:
Theorem 21.5 (Collapsing theorem). Let V' be a smooth projective surface
over a field к and let W' С V1 be an irreducible effective divisor on V!, which
as a scheme is isomorphic to P^, where К/к is some finite extension. In addi-
tion, let one of the two following equivalent conditions be satisfied'.
(i) T/l '2 Iц/' is isomorphic to 0(1) on P^, (here f C 0w is the sheaf of '
ideals defining W').
(ii)(K W') = ^[K: к].
Then there exists a birational morphism f: Vr V which is a monoidal trans-
formation with as centre the regular closed point x = /(И/') G V\ moreover, it
induces an isomorphism К^к(х). The morphism f which collapses W’ is
uniquely defined up to isomorphism. The surface V is projective.
Proof (outline). We restrict ourselves to showing how to construct the mor-
phism/which collapses W’. For the verification of its nice properties, see the
lectures of Safarevic [1] or, for surfaces over an algebraically closed field, the
treatise of M. Artin [1].
Let d - |X &]. Let L be a very ample sheaf on V1 such that the following
conditions are satisfied:
Я1(Г,Л) = 0,
(L, 0v,(W')) = ad. a€Z :
(As L one can take, for example, a sufficiently high power of an arbitrary ini-
tial very ample sheaf.) We consider the standard exact sequence
о-*0г(-аИ/')->0г->0аМ/,->о,
and take its tensor product with the invertible sheaf L ®Qy, OyiaW). The
exactness is maintained under this, and the third term remains as before. In-
deed, the sheaf L ® Оу.Оу^аМ'), restricted to aW’> has degree
(L, 0yf(aW)) + (aW*, aWl)~ a2d - a2d ~ 0 .
This restriction is therefore isomorphic to O^t because the invertible sheaves
on P1 are defined up to isomorphism by their degree (this same argument shows
the equivalence of conditions (i) and (ii) of the theorem). Therefore we obtain
an exact sequence of sheaves
O^L^L 0v,(aW’)^0aWI->0,
which gives an exact sequence of groups of sections:
Let Sq G£f°(F', L ®qv>be the inverse image of 1 G//°(K, 0aw),
and let Sp ..., sn E i(H®(V’, £)) be a basis of this space. The £-morphism
f: defined by the sheaf L ® q y> 0 y(aW') and its set of sections
(Sq, ..., s„) is birational because it is an isomorphism with f(V) outside W'.
Moreover, for every x E we have Sq(x) Ф 0, ^(x) = ... = sn(x) = 0, so that
the whole divisor W' is collapsed into the point (1,0,..., 0) G P£.
This construction has the, following geometric meaning. Let V' E P™
from the very beginning, and let L ’ - 0(1) be the induced sheaf on Vr. Replace
Lr by L = L’€ such that the conditions mentioned in the beginning of the proof
are satisfied. Then the zeros of the sections of the sheaf £ on V' are geometric
sections of V’ with hypersurfaces of degree e. The zeros of the sections of the
sheaf £ 0y'(aWf) form a linear system in which there is a linear sub-
system of codimension one consisting of hypersurface sections of degree
e plus ah7'. All these sections are zero on aWf and there is only one more inde-
pendent section $q, which is constant on aWr and which guarantees that there
is no ‘not being defined’ in the points of Wf.
21.6. The curves W’ on V’ which satisfy the conditions of Theorem 21.5 are
called exceptional curves. (In an older terminology, ‘irreducible exceptional
curves of the first kind’, in contrast with reducible curves which can be coL
lapsed by a birational morphism on the one hand, and on the other hand,
curves ‘of the second kind’ which can be collapsed under some birational map
with points on this curve where the mapTsmoLdefined).
Combining Theorems 21.4 and 21.5 yields the following important concept:
Definition 21.7. Let К be a smooth projective surface over a field k. Then
V is called minimal if one of the following two equivalent conditions is ful-
filled:
(i) Алу birational й-morphism/: V-> V', where Vr is a smooth projective
surface over k, is an isomorphism (i.e., it is impossible to shrink K).
(ii) There are no exceptional curves on V.
We immediately remark that minimality can be destroyed by an extension
of the field of constants. This circumstance will repeatedly arise further on
because it is characteristic for surfaces which are biration ally equivalent to
P2 over the closure of the base field. In order to follow this phenomenon, we
shall use the following criterion for minimality:
Theorem 21.8 (Minimality theorem). Let V be a smooth projective surface
over a perfect field k, and let G = Gal (k/k) (the Galois group of the algebraic
closure of the field k). We define the natural action of G on the group
Div (V ® k) (by the action on the field of constants к and the trivial action
on V). The surface V is minimal if and only if for every exceptional curve D
on V ® к there exists an element sEG such that s(D) and the intersec-
tion D О s(D) is non-empty.
Proof. In fact, suppose first that К is not minimal. Then there exists an
exceptional curveD on V. It is easy to see that D® к on V® к splits as a
disjoint union of a finite number of exceptional curves which are conjugate
to each other. This follows from the fact that D is the inverse image of some
closed point x with respect to some monoidal transformation f: V-+V', and
then / ® к : V ® к -> V' ® к will be a monoidal transformation with as
its centre the subscheme x® к which splits up as a union of deg x= [&(x): k]
conjugate closed points:
d d
x®k = Uxit D® к = U (•*,)•
z-1
None of the components of the curve D® к satisfies the condition of the
theorem.
Conversely, let Dy be an exceptional curve on V® к such that all its con-
jugates Dy, ... , D$ are pairwise disjoint. We shall first of all show thatZ>' =
- comes from some divisorD on У: Dr ~D ® к. This is a particular
case of a useful general lemma on ‘Galois descent’.
Lemma 21.8.1. Let V be an irreducible variety over a field k,K3ka
Galois extension with group G. The natural map Div V Div (К ® K):
D ® К identifies Div V with the subgroup of G-invariant divisors
(D\n(V®K))g.
Proof. First of all, the map D D ® К is injective and its image belongs to
(Div(K® A?))G. Conversely, we shall show that any elementZ>'€(Div(K®A’))G!
comes from a divisor on И If suffices to consider the case when ,
where the Z>2- are effective and irreducible divisors over К which are conjugate
to one another (over k) and pairwise different. Indeed, such divisors constitute
a free system of generators for the group (Div(K ® K))G.
As every divisor on V ® К is defined over some finite extension of the
field k, we can assume that К/к is finite. Let [A?: fc] = nd, We show first that
the divisor nD’ comes from V. In fact, let x G К be an arbitrary point, Ub x
an affine neighborhood such that the divisor Dy is given by one local equa-
tion ty in U ® K, Then t ~ ) *s a local equation in U ® К for the
divisor nD'. However, t G Г (U ® K, Oy 0 g)G= F(t/, Oy), and therefore
nD' ~ D ® K, where D is given over U by the equation t, (It is clear that when
multiplying ty by an invertible element t is also multiplied by an invertible
element, so that the local equations for D are compatible.) Now we must show
that if an effective divisor D G Div V can be divided by an integer m in
Div ( V ®K), then this can be done already in Div V. It suffices to verify that if
D is irreducible, effective and not divisible by any integer > 1 in Div V, then
it is also not divisible in Div(F ® K), We consider D as a closed subscheme
in V. Then the fact that it is not divisible in Div V implies that there are no
nilpotents in the structure sheaf Op. But because the extension К D к is sep-
arable, there are also no nilpotents in the sheaf Op q & = Op ® K. This
concludes the proof of the lemma.
21.8.2. We return to Theorem 21.8. We recall that an exceptional curve
Dy on V ® к is under consideration and that , D^ are the other
conjugates ofD, Let ^-yD. = D ® k, D G Div V. We shall show that/) is an
exceptional curve on К We exploit the criteria of Theorem 21.5. Clearly,
(/), /)) = -d.
It is therefore sufficient to verify that -(-a-s-a-scheme over k)D is isomorphic to
P^, where К is the minimal field over which the curve Dy is defined (in other
words, К = kH 9 where H ={'s G GI s(/)j) ~ Dy}f
We introduce a А-scheme structure on D as follows. The group G acts on
the structure sheaf 0$®% (as quotient sheaf of and
where is concentrated on the component Di ® к CD ® k. The sheaf
= C ( ®
coincides with the image of the homomorphism
if G ~ $.77, Z), = SjDy. On the other hand, there is a natural sheaf of K-
algebra structures on (0^ £)# because kH = K. We now show that
the К-schemes D and P^ are isomorphic. They become isomorphic over
к because D ® к, as shown above, coincides with (/^ ® к)н ® к ~ Pl
since Dy is an exceptional curve. Therefore D is a form of the projective line.
It is known that a non-trivial form of the line over К has no А-points; on the
other hand, there is an element of degree 1 in PicZ> (induced by the conormal
sheaf to/)), and its non-zero section becomes zero in a rational Appoint on/).
Consequently,/) ~ P^, which concludes the proof of the minimality theorem.
Example 21.9. Minimal cubic surfaces. Let К be a smooth cubic surface
over a field к in P3. First of all we shall show that the exceptional curves D on
F® к = Fare precisely the lines lying on V ® k. In fact,/) = Pl, and there-
fore
2рд(/))-2 = (£2г, 0r(/))) + (0r(/)), 0r(/))) = -2 .
Because D is exceptional, (0^(/)), 0p(/))) = — 1. It follows from this that
But Hp1 = z *((?p3(l)), where i: V is the usual embedding (cf. Safarevi6
[2], Chapter III, § 5). Thus D as a curve in P| has degree 1, and hence is a
line. By repeating these arguments in inverse order we obtain that the
self-intersection number of every line D on V ® к is — 1, so that it is ex-
ceptional in virtue of the criteria of Theorem 21.5.
A classical result is that there are precisely 27 lines on V ® k, so that
V ® к can never be a minimal surface. In Chapter IV we shall prove this and
study in detail the symmetry properties of this configuration of lines. Mean-
while we restrict ourselves to the verification of the fact that the surface of
Example 16.3, for which all lines can be explicitly written out, is minimal.
Nothing prevents us to study a somewhat more general case. Let & be a
field of characteristic Ф 3 containing a primitive cubic root of unity 0, and
let a (k*)3. Let Lbe given by the equation
T3 + T3 + T3 + aT3 = 0 .
Every line on V ® к is described by one of the pairs of equations
where (z, /, I) run through (0, 1,2) such that i < j, 0<и 2 and a* is any
fixed root in k. The Galois group G only acts non-trivially on the coefficient a3
in these equations and sends a? into в a3 or 02a3. Therefore the collection of
all lines splits up into 9 triples of conjugate lines. The lines of every triple in-
tersect at one point and they lie in one plane defined over k. It follows from
Theorem 21.8 that V is minimal.
Exercise 21.10 (Segre [2]). Show that the surface V given by 2?=oafT?= 0
is not minimal if and only if (k*)^ for some permutation of
(z, /, I, ti) of th$Jndices (0, 1,2, 3). If this condition is fulfilled, then on V ® к
there is a triple of conjugate lines which are pairwise disjoint.
22. Bibliographical remarks
The whole theory of this chapter orginated in the classical Italian school.
Contemporary accounts can be found in Zariski [1]. in the lectures of
Safarevid [1] (the theorem on resolutions is also proved in the lectures [2]),
in the treatise of Lipman [ 1 ], and-in-the-book ‘Algebraic Surfaces'. This book
also contains a description of all minimal rational surfaces over closed fields;
for more expositions see the treatises of Nagata [1], Hartshorne [1] and also
Manin [3], where the classification of minimal surfaces with a group of oper-
ators is obtained.
CHAPTER IV
THE TWENTY-SEVEN LINES
23. A survey of the results
Whole books have been devoted to the configuration of the 27 lines on
a smooth cubic surface (Henderson [1]; Segre [2]). Their elegant symmetry
both enthrals and at the same time irritates; what use is it to know, for in-
stance, the number of coplanar triples of such lines (forty five) or the num-
ber of double Schlaffli sixfolds (thirty six)? The answer to this rhetorical
question is one of the two recurring themes of this chapter. In just a few
words: the classes of the lines on V ® к generate the group 7V(K) =
- Pic (V ® к ) and the action of the Galois group G ~ Gal (k/k) on N(V)
preserves symmetry and it implicitly contains an extremely large amount of
information on the arithmetic and geometry of V.
Here is a short list of valuable results.
Theorem 23.1. Let к be a finite field of q elements, F G G the Frobenius
automorphism z z^, and let F* : A?(K) 7V(L) be its action on the
' Picard group. Let N denote the number of к-points of the surface V. Then
N = q2 + q Tr F* + 1 .
This theorem is due to Weil [3]. It is true for any smooth projective sur-
face over A: which becomes birationally trivial over k.
Now let A; be a global field, that is, [A:: Q] < °°. Let x denote the char-
acter of the representation of G on A(T), and let L (s, x, k) be the Artin
function of the field к corresponding to the character x (cf. Serre [5]).
Theorem 23.2. The Hasse—Weil zeta function of the surface V coincides
(up to a finite number of Euler factors) with the product
s(s, k)S(s-2,k)L(s-l,xIfc)
This result is also due to Weil.
Up till now we have had no means of determining whether a surface V
is birationally non-trivial over k, whether it can be trivialised over к or
whether Vfk) is non-empty. The G-module 7V(K) itself is, generally speaking,
not a birational invariant (cf., however, Theorem 33.2 of Chapter V), but
on the other hand its first cohomology group is invariant.
Theorem 23.3. (i) The group (G, N(V)) is a birational invariant of
the surface V\ifV = P2, then H\G, 7V(7)) = 0.
(ii) Suppose that the surface К is unirational. Then the degree of every
map P2 V over к is divisible by the exponent of the group H\G,
Example 23.4. It is impossible to describe all rational solutions of the
equation Tq + T^ + T^~ 2 by means of rational functions in two indepen-
dent parameters. Let, in general, the field к contain a primitive cubic root
of unity, let a € k* and let Va be given by the equation’
+ Г3 + Г 3 + a Tf = 0 .
viz Э
If a G (A:*)3, then Va is birationally trivial. But if a $ (Л*)3, then
H4G, N(Va)) s Z3 X Z3 .
(cf. Table 1 in § 31, or the direct calculation in Chapter VI).
Theorem 23.3 shows that the degree of unirationality of V is not less
than three. In Chapter VI we shall see that in this case no finite number
of two-parameter families of points can exhaust all points (over a number
field). Even more, except for the trivial case a G 6(k*)3, the surfaces Va and
cannot be birationally equivalent over k. This will follow in full gener- >
ality from the results of Chapter V. The cohomological invariant permits
us only to establish that Va and are not birationally equivalent over к
if the fields k{(T) and k(Z^) do not coincide, that is, if a (V)3. In
fact, in that case
Z3XZ3 ^(Gal^/fc^1)),^ ® £(O)
^(Gal^tM)), ® &(M)))M0}.
Informally speaking, this means that the problems of representing different
numbers as a sum of three cubes are independent.
23.5. The group if (G, which here emerges ad hoc, acquires an in-
teresting interpretation in Chapter VI in terms of Azumaya- Grothendieck
‘sheaves of simple algebras’. Using this, we then shall be able to construct other
useful things, including the following:
(i) A new admissible equivalence relation on Vfkf eminently calculable,
with an Abelian CML.
. (ii) An obstruction to the Hasse principle which explains all counter-
examples known in the literature.
Returning to the survey proper we conclude our list with the following
criterion for minimality (Segre [-1 ]).
Theorem 23.6. A smooth cubic surface V over к is minimal if and only if
N(y~)G = Zuyt where ccy is the class of the sheaf of differentials £2^.
All this sufficiently justifies our interest in the action of G <yn.N(V). This
action preserves a>y €N(V) and the intersection number; this permits a long
digression from algebraic geometry for the sake of the combinatorics of root
systems and their Weyl groups. This is the second theme pervading the present
chapter.
Definition 23.7. Let r > 1 be an integer. We consider a composed object
{Nr, ыг, ( , )}, where
(i) Nr = Z''+1 = ®/=Q ZZ;-; (/,) a chosen basis.
(ii)ay = (-3.1,...,
(iii) (,) is a bilinear form Nr X -> Z given by the formulae
(Zo,Zoj=l, = , (//,//) = 0ifi#=7.
We define the subsetsRr, Ir CNr by the conditions
(iv) Rr = {I €Nr I (Z, w,.) = 0, (Z, Z) = -2}.
(v) Ir = {l&Nr\(I, u>r) = (Z, Z) = -1}.
The main justification for these constructions is the following.
Theorem 23.8. Let V be a smooth cubic surface over a perfect field k.
(i) The triple {7V(K), coy, intersection number} is isomorphic to the ob-
ject , co6, ( , )} described in Definition23.7.
(ii) The Galois group G acts on N{V) and preserves Ыу and the intersec-
tion number,
(iii) The set of classes of lines on V 0 к goes over into Ir under the iso-
morphism (i).
Remark 23.8.1. It is useful to consider all values of r,.and not only r = 6, for
two reasons: Firstly, because induction on r is often convenient in the proofs.
Secondly, and most important, the algebraic—geometric objects, the Del Pezzo
surfaces, can be connected with all values 0 < r < 8 and not only with r - 6.
The Del Pezzo surfaces with r C 5 are obtained as images of non-minimal
cubic surfaces under birational morphisms. The surfaces with r = 7 and r = 8
are even more interesting, but shall not be investigated in detail here. All
these points have been omitted in the survey, so as not to drown the main
themes, but they occupy a large place in the main text of this chapter.
Remark 23.8.2. In the formulation of Theorem 23.8, nothing has been
said about the role of the set Rr. It permits us to identify the symmetry group
of the lines with the Weyl group of type E^.
Theorem 23.9. Let 3 < r < 8.
(i) The scalar product {with opposite sign) on R 0 z Nr Rr+1 induces
on the orthogonal complement of <лг the structure of a Euclidean space. The
set Rr is a root system in it of type А X A 2, AE& E^, E%, respectively
{the sum of the indices is equal to r).
(ii) The following groups coincide'.
(a) the group of automorphisms of the lattice Nr preserving cor and the
scalar product',
(b) the group of permutations of the vectors from Ir preserving their
pairwise scalar products’,
(c) the Weyl group W{Rr) of the system Rr generated by the reflections
with respect to the roots.
Remark 23.10. Starting with r = 9, the system Rr beomes infinite. In
Chapter V, a basic role is played by some variant of the object {/V^, ( ,)}
which, however, is best investigated by not losing sight of its algebraic—geo-
metric origin. ...................
23.11. In principle, Theorems 23.8 and 23.9 permit us to give all information
on the action of G = Gal (k/k) on7V(K) a tabular character. More precisely, the
representation of G inTV(K) is described by a pair consisting of the invariant
subfield of the kernel of this representation (a finite extension of k) and the
image of G in the group AutTV(K); this image ‘is’ a class of conjugate subgroups
in h'(£’6). Its invariants constitute the formal part of the picture. Therefore a
good table must contain a list of the classes of conjugate subgroups H in W(Rr)
and for every class the values of at least the following invariants:
(а)Я1 (Hf Nr) (Theorem 23.3);
(b) the character of the representation (Theorems 23.1 and 23.2);
(c) the multiplicity of the identity representation (Theorem 23.6);
(d) the decomposition of Ir into Я-orbits (this is especially useful for the cal-
culations of Chapter VI).
Some information of this type for the values r - 6 and r = 5 can be found in
the last section of this chapter. For r = 7 and 8, it would be reasonable to at-
tempt a computer calculation.
23.12. The realisations of the systems/^, described in this chapter
in the form of Rr are different from the standard ones (cf. for example Serre
[4] or Bourbaki [1]) and in a number of respects they are more convenient.
One of the advantages is that in the space Rr+1, which has only one dimension
more than the rank of the system, together with Rz there is also room for the
system of ‘exceptional vectors’ Ir. This can facilitate, for instance, the com-
binatorial part of the theory of exceptional simple groups over a non-closed
field.
23.13. Realization problems. We have in mind the following group of prob-
lems: Let be a perfect field, and let V run through the smooth cubic curfaces
over fc
(i) Which subgroups of М-^б) are rea^se^ as an image of the Galois group
by its representation on Pic (F® k) for some VI
(ii) Which extensions of the field к correspond with kernels of such repre-
sentations?
Of course, the answer must depend heavily on k. The derived groupW'(#6)
is simple, and the whole of it obviously cannot be realized over a local base field
which has only solvable extensions. On the other handy according to Todd [1]
and Segre [1], W(E6) can be realized over some function field. The proof rests on
Lefshetz’ theory: for K, one takes a general fibre of the bundle of sections
of a 3-dimensional cubic hypersurface. The local monodromy map for a
suitable choice of V generates all of PF(E6). The more interesting case of a
number field is completely unknown. The search for a reasonable approach
to problem (ii) may involve a pattern for a non-commutative class field theory.
Here, it appears that topological-analytical considerations and something like
Hecke operators do not suffice. The lines on a 3-dimensional cubic, connected
with Abelian varieties, appear to be still more promising.
The final problem is:
(iii) To what extent does the G-module 7V(K) determine the surface K?
It is probable that this problem only becomes reasonable for minimal cubic sur-
faces. An analogy with Tate’s theorem for Abelian varieties permits us to hope
for an interesting answer.
24. Del Pezzo surfaces
In this section we suppose the base field к to be algebraically closed. We
shall introduce and study a certain class of surfaces: they are smooth and pro-
jective surfaces over к and are singled out by simple invariant properties. From
the point of view of birational geometry, one naturally considers the surfaces
of this class simultaneously. We start with cubic surfaces.
Theorem 24.1. Let V be a smooth cubic surface over a field к. Them
(i) V is birationally trivial.
(ii) The anticanonical sheaf £2^ is ample. More precisely, flp1 = ^jzO)
under the usual projective embedding. .
Proof. In virtue of Theorem 12.11, Kis unirational. It follows that V is bira-
tionally trivial over closed fields, for instance, by means of the rationality cri-
terion (see “Algebraic Surfaces'", Chapter III, or Serre [1]).
Another, more elementary argument uses the fact that there exists a
line/) on V (Safarevic, lectures [2], p. 59). Taking the bundle of planes
through this line, we obtain a bundle of conics on V complementary to D.
Writing down the equations of V and D explicitly, one can convince oneself
that precisely five of these conics in turn decompose int o a pair of lines.
Choose one of these and repeat the process. This furnishes a line (the twelfth)
Dr on V which crosses the first one. The rational map D X Df *-> V:
(x, j) ь» x о у establishes the birational equivalence of V with P1 X P1 as in
Example 12.6.
Finally, the second assertion is well known in a wider context, namely for
hypersurfaces (and complete intersections) of arbitrary dimension. It follows,
for instance, from the addition formula, if one takes into account that £2рз =
“0рз(~4) and Орз(К)= Орз(З). This proves the theorem.
Definition 24.2. A smooth birationally trivial surface V on which the sheaf
£2^ is ample is called a Del Pezzo surface.
We recall that the property of being ample by definition means that there
exists an integer n > 1 and a closed embedding i \ Vе-* such that Slyn =
= z *(0j^(l)). If one can take n = 1, then the sheaf £2^ is very ample. The
number d = (coF, coy) is called the degree of the Del Pezzo surface V. It coin-
cides with the projective degree of the image z(7) if co0 ~ z*(0(l)).
Theorem 24.3. Let V be a Del Pezzo surface of degree d. Then
(i) 1 <d<9.
(ii) Every irreducible curve with a negative self-intersection number on V
is exceptional.
(iii) If V has no exceptional curves, then either d-9 and V is isomorphic
to P2, or d ~ 8 and V is isomorphic to P1 X P1.
Proof. Assertion (i) is immediately implied by the following lemma if we
take into account that (£2^, £20 > 1 because £2^ is ample, and that
rk Pic V > 1.
Lemma 24.3.1. For every {smooth projective) surface V which is biratio-
nally trivial, the group Pic V is free with a finite number of generators and
rk Pic K+(£2^, £20 = 10 .
Proof. Let Vr -> V be a monoidal transformation with a closed point as
centre. It follows directly from the results of Section 20 (particularly from
Corollary 20.9.1 and Proposition"2Orf0)~thaLthe lemma is true for V if and
only if it is true for V\ The resolution Theorems 21.1 and 21.4 then show
that it suffices to verify the lemma for a single arbitrary surface, for instance
for P2. In this case, the lemma holds because = 0p2(~3) and Pic P2 ~ Z.
This proves the lemma.
24.3.2. Proof of Theorem 24.3 (continued), (ii) Let DC Г be an irreduc-
ible curve and (D, D) < 0. Because Пр1 is ample, (D, ) > 0 (this is the
degree of the curve D, which divides n, if induces a projective embedding
of V and/)). On the other hand,
2pa(£>)-2 = (D,Z»)-(D>£2-1).
Butpfl(£>) > 0, asD is irreducible. The only possibility is therefore the case
determined by the equalities
(D,D) = -1, Pa(O) = 0.
The latter condition is only possible whenD ™ P1. The curve/) is therefore
exceptional (cf. Theorem 21.5 and Section 21.6).
(iii) . If there are no exceptional curves on V, then V is minimal and by (ii)
it has no curves with a negative self-intersection number. But except for P2
and PW, there do not exist minimal rational surfaces with this property
(Section 18.7.3). On the other hand, they are both obviously Del Pezzo sur-
faces. This proves the theorem.
By Theorem 24.1, a cubic surface is a Del Pezzo surface of degree 3. The
reason for considering all Del Pezzo surfaces is that this class is closed under
ordinary birational morphisms (cf. Corollary 24.5.2). The following theorem
gives the main geometric information on these surfaces.
Theorem 24.4. Let V be a Del Pezzo surface of degree d.
(i) If d = 9, then V is isomorphic to P2.
(ii) If d =8, then V is isomorphic to either P1 X P1 or to an image of Pl-
under a monoidal transformation with its centre at one point.
(iii) If 7 > rZ > 1, then V is isomorphic to an image ofP^ under a monoidal
transformation with as centre the union of 9 —d closed points, no three of
which lie on one line and no six of which lie on one conic.
Conversely, any surface described under (i), (ii),(iii)/<9rd d? 3 is a Del
Pezzo surface of the corresponding degree.
Remark 24.4.1. For 7 > d > 5, all Del Pezzo surfaces of the same degree
are isomorphic. In fact, the projective automorphisms of P2 act transitively
on systems of < 4 points in general position.
Remark 24.4.2. The last assertion of Theorem 24.4 should remain true also
for d ~ 1,2 if the requirement on the ‘general position’ of the blown up points
is strengthened (see Theorem 26.2 for this approach).
24.4.3. Proof of Theorem 24.4. The minimal V have already been described
in Theorem 24.3 (iii). Let Fbe non-minimal. Then there exists a birational mor-
phism f: V -+ W, where W is a minimal rational surface. JV cannot be a non-
trivial ruled surface; otherwise there would be an irreducible curve D on W
with self-intersection number —2; then (f^(p),f^(pj) — 2, which con-
tradicts Theorem 24.3 (ii). Therefore either W = P2 or W - P1 X P1. In the
second case, let x E W be a point where f~l is not defined. According to
Lemma 21.4.1, the morphism/can be split up into morphisms V Д. W'-+ W,
where W' -> W is a monoidal transformation with centre x. But for W' in turn
there exists a morphism h : Wf P2 collapsing the inverse images of the two
fibres of the projections of W on P1 which pass through x (cf. e.g. Example
21.3). The composed morphism V Л W' Д>Р2 gives also in this case a bira-
tional morphism of V to P2.
We shall, as before, denote this morphism by/. Since under each monoidal
transformation with a closed point as centre, the rank of the Picard group goes
up by one, and the rank of this group for V is 10 — d (Lemma 24.3.1), /splits
up as a product of r = 9 — d of such transformations. In other words, the in-
dex of/is equal to r in the terminology of Definition 18.1.2.
Let xj,..., xs E P2 be all closed points in which /“^ is not defined. Then
s = r necessarily. Otherwise, s < r, and then one of the monoidal transformations
of the decomposition of /would have its centre on the inverse image of some
point xz-, under the blowing up of this point. After that transformation, the
proper inverse image of D (Definition 20.3) would have an intersection number
—2, and on V this number could only diminish still more. This contradicts
Theorem 24.3 (ii).
Suppose that three of the points Xp..., xr G P2 are on one line D, Then
— 2 because the self-intersection number becomes —2
already after only blowing up those-three points. Analogously, a conic on
which there are six of the points Xp ..., xr is converted into a curve with
self-intersection number < — 2. All this is impossible according to Theorem
24.3 (ii). .
Finally, we prove the last assertion. It follows from the much stronger fact:
Theorem 24.5. If the surface V is obtained from the plane P2 by means of
a monoidal transformation with as centre r <6 closed points no three of which
are on one line and no six on one conic, then the sheaf eSy is very ample and
its sections yield a closed embedding of V in a projective space of dimension
dimtf°(K, Пр1) - 1 = (J2r,S2r) = 9 - r .
The set of exceptional curves is identified under this embedding with the set
of lines in the containing space which lie on V, The image of V has degree
9-r.
Proof. We divide the argument into a series of steps. Let Xp . .., xr G P2
be the points mentioned in the theorem.
First of all, we put£ = {s G7/° (P2, ^2~j) I ~ 0 for all /}. Because $2^ ~
~0р2(3),£ can be identified in any projective coordinate system with the
space of cubic forms which are zero in Xp ... , xr. In particular, dim/, >10 —r.
We now define a rational map
y.pS^pdiml-l =proj ф SKL ,
№=0
which lets a point x G P2(fc) correspond with the point with homogeneous
coordinates (sj(x),..., Sjiny/,(*))> X ($/) is a basis for L (because the
are sections of an invertible sheaf, their ‘values’ at a point x are defined up to
a common constant factor which can be fixed by choosing an isomorphism г
of the fibres -> 0Y).
(ar) The map f is defined everywhere outside \Л?=1х{; moreover, if Фу2,
thenf(yl)^f(y2).
We use induction on r.
The first assertion of (ar) follows from (arl). Indeed, let у but/not
defined in r.This means that all cubic curves passing through Xj. ,A'r auto-
matically also pass through v. Denote by L' the space of cubic forms which are
zero in Xp... ,xr_p In virtue of (a^), there exists an s ElL such that s(x.f.)E=0,
It follows from our assumptions that the ratio s(xr) : s(y) as a point in P1 re-
mains constant. (In fact, if s^x^): : 52^)’ ^ien there would be
some linear combination of and s2 which is zero in xr but not in у, that is,
/would be defined iny.) But then the map /' corresponding toxp..., xr_|
would not distinguish xr and y, which contradicts Д_Д
To prove the second assertion of (аД we choose first of all two points y^
and j>2 outside ЦД xz-} and we suppose that/(yj) “/(уД suffices to lead
this assumption to a contradiction in the case where r is equal to its maximal
possible value, six. In fact, if r < 6, we add to Xp . ..., xr the missing number
of points Tn sufficiently general position’; the contradiction obtained for the
new system is one for the old system too. Thus, let r = 6. We denote by D the
line through jq, y2; let D also contain 2 —к points of (xp... , x6), where
к ~ 0, 1,2. On // we choose к additional different points y2+z different from
Xp... ,x6,Tpy2.
The space of cubic forms is ten-dimensional. Therefore dim L > 4. Let
L’ = {s EL I = s(^2) “ OK 1^5 Is zero in one °f the pointsy^ y2, then it
is also zero in the other, because/(j^) =/(^2) (CK ^e arguments used above).
Consequently, dim L' > 3. Let L" ~ bEL' I ^(y2+y) = 0, i > 0} (L” ~ L'
if к = 0). Then dim Z" > 3 — к > 1. On the other hand, for 0 Ф sEL", the
cubic curve 5 = 0 intersects the line/) at the pointsyp j*2 and also at two
other points: those 2 — к which are contained among the (xp ... ,хД and the
к additional pointsy2+z. Therefore D is a component of the curve 5 = 0. Let
Q be the complementary component, a conic. For к = 2, it must pass through
all points xlr. . . , x6, which is a contradiction. For к = 1, it passes through
five of these points, missing, generally speaking, the one which lies on D. But
then dim L " > 2, and the conics constructed in this manner form a bundle
and one of its elements also contains the missing point, which is again a con-
tradiction. Finally, for к =0 the conic Q passes only through four points of
Xp ... , x6, but dim L” > 3 and we can again construct a conic passing through
the two additional points. This concludes the proof of (ar).
(ЬД dimZ = 10- r.
This can be immediately obtained by induction from (аД In fact, we
need to prove that adding a point xr to the points Xp ... , xr_j diminishes
the dimension of L (it follows that it then diminishes by one). If this were
not so, then the map/' corresponding to ..., would not be defined
inxr. This proves our assertion?------------
Now denote by V Яр2 the monoidal transformation with centre U.G
g э f о w 1
and let i: V P2 Д.Р9~Г be the composed rational map. The aim of the fol-
lowing is to show that i is the morphism of a closed embedding. This will be
achieved by a local analysis of f and g, but first we check that this implies the
main assertion of the theorem on the ampleness of ft/1 and the dimension of
the space of sections of this sheaf.
Indeed, first of all we have g*(ft~2 ) = ftp1 0 where D. is the
inverse image ofx7- on V (cf. Proposition 20.10). Further, #* defines an em-
bedding of L in the space of those sections of Я°(К, ftp1 0 0(S/= j £>/)) of
which the zeros are contained in Lf. But the space of sections of 0(S^| Dj)
is one-dimensional. Dividing by a generator of it, we can identify/, with a sub-
space in ftp1). Indeed, no other sections except those which come
from L are contained in ftp1): the projection into P2 of the zero divisor
of such a section must coincide with the zero divisor of some section of L
because (Z>z-, ftp1) = 1, and this means that (s = 0) intersects all the Conse-
quently, dim H®(V, ftp1) - dim L - 10 — r.
Finally, if we assume that it is known that i is a closed embedding, we ob-
tain immediately from its definition that ftp) ~ z*(0p9„r (1)). Indeed, the
linear systems of the zeros of the sections of both sheaves coincide on И
We now pass to the proof that i is a closed embedding. We shall employ the
following criterion. Suppose that it is already known that the map i is a mor-
phism at a point x EV Then it is a closed embedding locally in a neighbour-
hood of x if i induces a local isomorphism of V and z(K) in neighbourhoods
of x and z(x), respectively (because z(K) is closed in virtue of the fact that V
is projective). In turn, to verify this property, it suffices to know that the dif-
ferential^ : Tx -» has no kernel, where Tx is the tangent space at the
point x,that is, the space of linear forms on тх/т^ (where mx C Qx is the max-
imal ideal). In fact, it follows from this that the dual map ls
epimorphic, so that, by Nakayama’s lemma, z* : ls epimorphic.
It is therefore an isomorphism because z(K) is two-dimensional.
We first apply this criterion to those points у E V in which g is a local isomor-
phism. By (ar), z is a morphism in all such points. For convenience we can
assume that у E P2 x x^ and consider f instead of i.
(c).Ker^/=O.
Indeed, suppose that this is not true. As above, we can assume that r = 6.
Then, using the notations of the beginning of the proof of (ar),the zero divisors
of all sections 5 E L which pass through у either have at least a double point
at у or they all have a common tangent. In both cases, there exists a line
D С P2 for which the local intersection number with every curve (s = 0),
s EEL, is not less than two. Let D also contain 2 — к points of (xp ..., x6),
к - 0,1,2. OnZ> choose к additional separate points (y2) different from
Xp ... , x6, y. After this, the proof of (a6) can be carried over verbatim to
this case. (An intuitive explanation is that the pointsyj andy2 from (аб) ^ave
‘fused’ iny, and D is the limit position of the line through у j andy2.)
(d) . The rational map i is a morphism in all points of V which lie over
somexui к = 1,.... 6.
д. J J 7
If i is not a morphism in some such pointy € E, then the proper inverse
image on V of the zero divisors of the sections sEL must all pass through
this point. Let g(y) coincide with, let us say, x^; then all curves 5 = 0 must
have a common tangent atxp the direction of which corresponds withy. We
show that this cannot be the case. To this end, we consider a reducible cubic
curve consisting of a conic Q which passes through Xp x2, x3, x4, and a line
D passing through x5, x6. In addition we can choose the conic Q such that its
tangent atXj is any prescribed line. Indeed, take a system of coordinates in
P2 such thatxpx2, x3, x4 become (1, 0, 0), (0, 1, 0), (0, 0, 1), (1,1, 1). The
equation of Q then has the form a Ty T2 + b Tq T2 + c Tq Ty =0, and the line
cTy+bT2=0is its tangent at Xp (If c = 0 or b = 0, then Q decomposes, but
this does not interfere with the argument.)
(e) . Ker dy i = 0 in any pointy € V.
Again letg(y) =Xp We denote by Df the line on P2 through xt in the
direction corresponding toy. Suppose that we have found two curves (s = 0),
s E L, non-singular inXj, which are tangent to D' at x^ and such that their
order of tangency to each other is one. Then their proper inverse images on V
go through у in different directions. It follows easily from this that the kernel
of dyi at the point у is trivial. We now show how to find such curves.
Except forxj, there is on/У at most one more point of (Xp . .. , x6). Let
x2, x3, x4, x5 be not on£f We now apply the construction of (d) twice to
get conics which are tangent to/У at x^ through (xp x2, x3, x4) and through
(xp x2, x3, x5). These two conics, together with complementary lines, are
the curves we are looking for. Each of them is irreducible because the tangent
D* contains no points of the quadruple other than Xp they do not coincide
unless the conic through Xp x2, x$, x4, x5 happens to have D’ as tangent line
atxp In that case take the conic through Xp x2, x3, x4, x5 and the conic through
Xp x2, x3, z (where z is any sMBcierffly^pnefaT^mt) with tangent Dr at Xp
In both cases the order of tangency atx* of the curves is not larger than one
because otherwise their intersection number at this point would be > 3, and
they already have two other points of intersection: x2 andx3.
Finally, let D С К be an exceptional curve. Because (D, D) — (D, Sip1) =
= 2pa (D) — 2 ~ — 2 and (D, Z>) = — 1, we have (D, Sip1) = 1. This means that
the degree of D in the anticanonical embedding of Vis equal to one, that is,
D is a line. This argument can clearly be reversed. This concludes the proof
of Theorem 24.5.
Remark 24.5.1. Theorem 24.5 ceases to be true for r = 7,8. The critical
part is of course the end of the proof of (ar): the dimension of the space L"
does not suffice. Indeed, for r = 7, the map defined by the sections of 2^ has
degree 2: any cubic curve which passes through Xp ... , x7 and the point y^
necessarily also passes through some point у 2. F°r r = 8, for the same reason,
the one-dimensional space of sections of Sip1 has a base point; its resolution
constitutes a bundle of curves of genus 1, parametrized by the projective line.
Corollary 24.5.2. Let V' ->Vbe any birational morphism.
(i) If Vr is a Del Pezzo surface, then V is also a Del Pezzo surface.
(ii) If V is a Del Pezzo surface, rk Pic Vf < 7, and all curves on V’ with a
negative self-intersection number are exceptional, then V’ is also a Del Pezzo
surface.
Remark 24.5.3. The restriction on the rank in (ii) should not be there, but
I do not know how to get rid of it.
Proof, (i). We use the following theorem of Moisezon and Nakai (cf.
Cartier [1]): the sheaf Sip1 is ample if and only if we have (Op{D), Sip1) > 0
for all irreducible curves D. [It is clear that this condition is necessary. The
sufficiency is a much deeper fact, but geometrically it is plausible, for in-
stance, because the restriction of Sip1 to any curve, being a sheaf of positive
degree, is obviously ample.]
Now let D' C V', D ~f(Df) С V. Proceeding by induction on the index of
f, we can reduce to the case where/is a monoidal transformation with a point
x 6 V as centre. Let e be the multiplicity of x on D. Then/У~f*(D) — eW’,
where Wf is the inverse image of x. Moreover, (£)', £2^) > 0 because £2^ is
ample. But
(D, Q’1) = = (D'+ eW', aft) = (Б; fipl) + e > 0
(taking into account that (/'*(£)), 0(H'1)) = 0). Therefore, is an ample
sheaf and К is a Del Pezzo surface.
(ii). Now let К be a Del Pezzo surface. As in (i), we can assume that /is a
monoidal transformation with as centre a point x G V, If V~ P2, then the asser-
tion immediately follows from Theorem 24.5; for V= P1 X P1 it also follows
from Theorem 24.5, using the ‘stereographic projection’ as in the proof of
Theorem 24.4. If К is not minimal, let V Л P2 be its morphism, which exists
according to the proof of Theorem 24.4. At the point x G K, g is a local iso-
morphism; otherwise x would lie on one of the curves collapsed under g and
then the proper inverse image of this curve on Vf would have self-intersection
number —2, which contradicts the hypothesis. Therefore the composition
V' X V Л P2 is a monoidal transformation with as centre a finite number of
points — the union of the centre of g and /о g(x). All the points thus chosen
are in ‘general position’ as described in Theorem 24.4 (iii): in the opposite case
there would be on V’ a curve with self-intersection number < —2. The last
statement of Theorem 24.4 then shows that V' is a Del Pezzo surface.
This result is useful for proofs ‘by induction on the degree’, as we shall see
in the following section.
25. The Picard group and root systems
This and the following sections are devoted to the proof of the results de-
scribed in Definition 23.7 and Theorems 23.8 and 23.9.
Proposition 25.1. Let V be a Del Pezzo surface of degree d which is not
isomorphic to P1 X P1; we put r ~ 9 — d. There exists in the Picard group
Pic Va free basis (Zq, Zj,..., lr) such that
(i) CO у - — 3/q + ly
(ii) (Zo, Zo) = 1, (//, Zz) = -1 for i > 1, and (f, Zy) = Ofor i ^j.
Corollary 25.1.1. The object {Pic К intersection number} is, up to iso-
morphism, only dependent on r and coincides with the object {Nr, cor,( , )}
described in Definition 23.7--------------7
25.1.2. Proof of Proposition 25.1. Let/: P2 be the monoidal transfor-
mation with a centre of r points (Theorem 24.4 (iii)). Let Zq be the class of
/*(0p2O))> and tet Jr(f-> 1) be the classes of the sheaves Oy (Pi) where the
Di are the inverse images of the blown up points. They constitute a basis for
Pic V according to Corollary 20.9.1. The formula for go у follows from Propo-
sition 20.10 if one takes into account that fZp2 s 0p2(—3). The self-intersec-
tion numbers are calculated by means of Lemma 20.9. These establish asser-
tions 23.8 (i) and (ii).
We shall now occupy ourselves with the system R?. We keep all the nota-
tions of Section 23.
Proposition 25.2. The orthogonal complement go? to cor in R ® N
equipped with the scalar product of Nr (with opposite sign), is a Euclidean
vector space of dimension r for r < 8. The vectors from Rr in it form a system
of roots of rank r.
Proof, (i). Obviously, the space R ® TV is generated by the vectors
Zj,..., lr. Moreover,
( Г \ Г
G0r, acor + Z/ I = (9 - r) a - Z/ .
z-l ' z=l
Therefore,
(r к r
Gor, acor + Z/ b^ j = 0 о (9 - r)a = Z/ \ .
z-1 ' z~T
The length of such a vector (with opposite sign) is equal to
so that the intersection number on is negative definite.
(ii) . We now check that the set Rr of vectors from Nr of length 2 and or-
thogonal to cor forms a system of roots. According to the definition (cf. Serre
[4], Ch. 5.2), we must establish the following facts:
(iia) Rr is finite, generates go± and does not contain zero. It is clear that
Nr A cof is a lattice in it can therefore contain only a finite number of
vectors of a given fixed length. Finally, the r vectors Zj - Z2,... > h ~ lr>
-l^- h are linearly independent and have length 2.
(iib) For every element I €Rr, the reflection of the space (flr,
s,(x) = x + (x, /) /,
sends Rr into itself. (Our formula for is obtained from the usual one after
a change of sign of the scalar product if one takes account of (I, I) ~ — 2.) This
is obvious because the symmetry S; preserves length and orthogonality to ,
and it sends Nr into itself.
(iic) 2(4 m)/(ltl)6Z for all Z, m ERf„ This is obvious.
This proves the proposition.
To identify the root systems R? which we have constructed, we establish
one of its invariance properties. In keeping with the notations of Bourbaki [1],
we consider the group Q(Rr) G cojt generated by Rr and we set
P(Rr) = {I E uA I (I, m) EZ for all m E 2(/?r)}
(the dual lattice). Clearly, Q(Rr) G P(Ar) and therefore the quotient group
P(Rr)/Q(Rr) is finite. Its order is called the ‘index of connectedness’ of the
system Rr.
Proposition 25.3. For all r, 3 < r < 8, we have
P(Rr)IQ(Rr) = Z9_r.
Proof. It is easy to check that the vectors Zj - Z2,.. . , l± - lr Zq -- Zj -Z2~Z3
of Rr generate Nr П ; therefore they form a basis of the lattice Q(Rfl.
We now consider the homomorphism
X:P(Ap^Q/Z,
for which
Г X
XI al$ + Z/ bjlf I - (mod Z) .
\ f=i /
Writing out the condition that the scalar product of the elements
with I! - Zz and Zo- l± -- Z3 must be an integer, we obtainby-bieZand
~b3 G Z; moreover,гЗя—= O (orthogonality to cor). It fol-
lows from this in the first place that Ker x = Q(fir) ~ And in the
second place,
a ~ 3bx G Z ,
3a-rb^ GZ .
These conditions are equivalent to what we want: (9—r)b^Z. From this one
infers that x induces an isomorphism
WWr)~ ^-z/z,
which is what we desired to prove.
As was remarked by P. Deligne, the properties thus established for the sys-
tem Rr suffice to identify it uniquely as one of the classical root systems:
Theorem 25.4. Let Rr for 3 8 be я root system of rank r with the
following properties',
(i) The lengths of the roots are the same.
(ii)P(Rr)/Q(Rr)^Z9_r.
Then Rr is isomorphic to one of the following systems, which are ordered ac-
cording to increasing rank*.
A^X Ay A^, D$, E& Ey E$ .
Proof. We turn to the tables in the book of Bourbaki [1], pp. 250—275.
The systems of type Cy and G2 do not satisfy condition (i). Of the re-
maining systems, only E% has its index of connectedness equal to 1, therefore
Ag =2Tg. The index of connectedness of a direct sum of root systems is equal
to the product of the indices of the summands. Therefore for 9 — r - 2, 3, 5
(primes) the systems Rr do not decompose into a direct sum. For these values
of the rank and the index, the tables directly give Rq—E^, R^E& R^—A^.
For r - 5, the system D5 is the only indecomposable one of index 4. If R5
were decomposable, then its two summands would have to be simple, of
index 2 and their ranks must sum to 5. But Dj has index 4 and Л; has index7+1,
so that it is impossible to get as a combination of these. One proceeds anal-
ogously to identify R 3 with A j ХЛ2. This proves the theorem.
We can now apply Theorem 25.4 in two directions: to obtain necessary
information on the groups Pic F, ‘anticipating results’ by looking in the
Bourbaki book, or to prove properties of Rr, using our new geometric realiza-
tion. We shall do both.
25.5.1 . Diophantine equations for Rr. Let
I ~ al® — X/ b-l-ENr .
zM
The condition I ERr means that the coefficients a, G Z satisfy the system
of equations
За - E bf = 0 ,
Z=1
(iv.i)
a2-tb^~2.
z=l
This interpretation is convenient in order to see that the Rr can be embedded
in each other as lattices,
r
R = Ra П. ® RZ. .
r 8 z
In particular, all calculations can be carried out in Ag, and for the remaining
Rr one simply ‘omits the unnessary’. (The system R^ =E& which corresponds
to a cubic surface, in this context does not particularly stand out in any respect.)
25.5.2 . The number of roots and an explicit description of them. It is easy
to check that the following table lists all integer solutions of the system (IV.I)
with the properties 0 <zz < 3, by > Z?2 > • * • ^8-
a bl b2 z>3 b4 b5 b6 Ьу b8
0 1 0 0 0 0 0 0 -1
1 1 1 1 0 0 0 0 0 (IV.2)
2 1 1 1 1 1 1 0 0
3 2 1 1 1 1 1 1 1
Proposition 25.5.3. A ll roots ofR% can be obtained from the roots (IV.2)
by reversing signs and permuting the b^ in all possible ways.
Proof. First method. We consider an arbitrary solution (a, by, ... , b8) of
the system (IV. 1). Permuting the bz- and changing signs if necessary, we can
assume that a > 0 and by > ... > b8. Then it follows from the first equation
of(IV.l)thata = |Z®=1b.<Z>i + b2 + b3-Let 1 = 0, b i, 1,0,0,0,0,0). The
reflection with respect to I sends (a, by,..., b8) into (zz+ c, by c, b3 +c,
b4,..., b8), where c = a - by — b2 - b3 < 0, and hence it diminishes a. There-
fore all roots can obviously be obtained from the table by applying to it not
only permutations of the bz and changes of sign, but also the reflection Sj. But
this reflection does not permit us to obtain a solution with Ы > 3 because
12a-bh — Ь^ - bj К 3 for all the tabulated solutions and their opposites. This
proves the statement.
Second method. We denote by Ry the set of solutions of the system
(IV. 1) with r - 9. We shall presently see that it is infinite and is not a root sys-
tem; A8 is obtained from this set by omitting all solutions for which b^ ¥= 0.
The introduction of the auxiliary unknown b9 is convenient because the equa-
tions (IV. 1) with r ~ 9 are equivalent to the equations
9
3a - 2/ by = 0 ,
, <!V-3)
Й(а-3»,)2 = 18 .
/=1
In total there exist three essentially different representations of 18 in the
form of a sum of 9 squares of numbers which are in the same residue class
mod 3:
18 = 32+ З2 + 02 +... + 02 = (±2)2 + (+2)2 + (+2)2 + (+l)2+ • •• +(+l)2 •
Therefore, up to the enumeration of the all solutions of (IV.3) are con-
tained in the following series:
(3Z>;Z>+l,Z>-l,d,...,d), (3Z>±2;Z>, Z?,Z>,d±l,...,Z>±l). (IV .4)
(b G Z; in the second series we must either take all lower or all upper signs.)
One easily sees that all solutions of (IV.4) for which one of the bj is equal to
zero can be obtained from the table by a permutation of the bt and a change
of sign.
The system will be useful in the following section for the description of
exceptional curves.
Corollary 25.5.4. The number of elements of Rr is given by the following
table:
8 7 6 5 4 3
----------------------------------------- (iy5)
240 126 72 40 20 8
Proof. A direct calculation using Proposition 25.5.3.
25.5.5 . Simple roots. The following proposition describes a system of simple
roots, or a basis in the terminology of Serre [4].
Proposition 25.5.6. (i) In terms of Table (IV.2), the following roots consti-
tute a basis S8 CR8:
(0,-1, 1,0,... ,0),(0,0,-1,1,0,...,0),(0,0,0-1, l,0,0,0),..., (0,..., 0, -1,1)
(1,1,1,1,0,0,0,0)
(They are arranged as in the traditional Coxeter-Dynkin diagram (see 25.5.7).)
(ii) The intersection Sr-S^ 0 (®/Q R/-) is a basis for Rr for allr^3.
Proof. We shall construct a basis for as described in Chapter 5, § 8 of
Serre [4]. We consider the linear form
8
(zz, by,. .., Z>8) a + Z/ ibi.
z-1
It takes on a positive value exactly for those roots which are obtained by per-
mutations of the bi of the first and third rows of Table (IV.2) and by those
permutations of the b^ of the first row for which —1 occurs earlier than 1.
The indecomposable ones of these positive roots are precisely those listed in
the statement of the proposition. The second assertion is immediately obtained
from this.
25.5.7. We enumerate the roots described in Proposition 25.5.6:
rl h r3 r5 r6 rl r8
We denote by the reflection with respect to rz-. Because sr(x) = x + (x, r)r,
we have the following explicit formulae:
SjCa, ftp ft2,..., ft8) = (a, b2, ЬуЬ3,...,Ь6)
s4^a> ft8) = (a + c, b{ +c, b2+e, b3 +c, b4,..., ft8) ,
c = a - by - Z>2 - &3 .
(s2 interchanges b3 and b%, s3 interchanges and b$, etc.)
25.5.8. The Weyl group Ж(АГ) is the group of permutations of the roots of
Rr generated by the reflections Sp ..., sr restricted to Ф^=о 1Ц.
All relations between the generators are consequences of the following:
1,
($ s.)2= 1 if r r are not joined in the Dynkin diagram ,
(sisf^ = 1 ifrj are j°ine(i
(cf. Bourbaki [1]).
26. Exceptional curves and Weyl groups
We keep the assumptions and notation of the previous section; in particular,
Kis a Del Pezzo surface of degree J, and r = 9 — d.
Let/) С К be an exceptional curve. Then the class I of the sheaf Oy(D) in
Pic V satisfies the condition
We shall call all such classes exceptional.
The isomorphism described in Proposition 25.1 of {Pic V, coy, ( , )} with
{Nr (x>r, ( , )} identifies the set of exceptional classes in Pic V with the subset
Ir C Nr described in Definition 23.7.
Let
(J I l Г
i=l
The condition lENr means that
3a - S b{ = 1 ,
(IV.6),
a2 - Zz Z>? = — 1 .
The problem of describing Ir for r < 8 is solved by means of two simple argu-
ments. Firstly,Ir = Zg fl (ф so that it suffices to calculate . Secondly,
the introduction in (IV.6)8 of an auxiliary ‘unknown’ = 1 immediately re-
duces the question to the solution of the system
9
3a — Zz Z>=0,
z-1
a2-Z/ft?=-2, b9= 1
Z~1 .
These solutions without the restriction = 1 are described in (IV.4). A simple
computation yields the following result:
(IV.7)
Proposition 26.1. All solutions of the system (IV.6)8 are obtained by all
possible permutations of the bj of the rows of the following table:
a b\ b2 b3 64 b5 b6 b2 b8
0 -1 0 0 0 0 0 0 0
1 1 1 0 0 0 0 0 0
2 1 1 1 1 1 0 0 0
3 2 1 1 1 1 1 1 о (IV-8)
4 2 2 2 1 1 1 1 1
5 2 2 2 2 2 2 1 1
6 3 2 2 2 2 2 2 2
We can now derive the main theorem on exceptional curves on Del Pezzo
surfaces.
Theorem 26.2. Let V be a Del Pezzo surface of degree 1 < d < 7, and let
f\ V P2 be its representation in the form of a monoidal transformation of
the plane with as centre the union of r ~9—d points Xp ... ,xr. Then the
following assertions hold'.
(i) The map D (class of Oy(D)) G Pic V establishes a one-one onto
correspondence between exceptional curves on Vand exceptional classes in
the Picard group. These classes generate the Picard group.
(ii) The image f(D) in P2 of an arbitrary exceptional curve D С V is of
one of the following types'.
(a) one of the points xy,
(b) a line passing through two of the points xy
(c) a conic passing through five of the points xy
(d) a cubic passing through seven of the points хг- such that one of them is
a double point;
(e) a quartic passing through eight of the points such that three of them
are double points',
(f) a quintic passing through eight of the points xi such that six of them
are double points.
(g) a sextic passing through eight of the points xfsuch that seven of them
are double points and one is a triple point.
(Of course, only for r~ 8 the whole list must be used', for r-’l only (a) — (d);
for r~ 6,5 omy (a) -- (c); for r = 4,3 и, „у (a) — (b).
(iii) The number of exceptional curves on V is given by the following table:
8 7 6 5 4 3
240 56 27 16 10 6
(IV.9)
Proof, (i). Let L be an exceptional sheaf on Vf that is,
(£,L) = (Z,Qf) = -1 .
Apply the Riemann-Roch theorem to it, take into account that pa(V)= 0,
discard//1 and replaceifi by its dual group. We then find:
dimfl°(F;Z.)>j(Z,Z)- £)+1 ® Z-1)
But
(Qp1,^ = <0.
Because the sheaf is ample, this means that f/jz ® L”1 has no non-zero
sections. Thus dim 1.
Let s be a non-zero section of L, and D the divisor of its zeros, and let
D " az- > 0, be its decomposition into irreducible components. Now
l=(np1,D) = Sa/(J2^Z>f),
and from the ampleness of it follows that (^Z1, Z>) > 0. D can therefore
have only one irreducible component which is of multiplicity one. From the
condition on L it follows that pa(D) = 0, (D,D) - —1; thereforeD is an excep-
tional curve and L s
Two different exceptional curves belong to different exceptional classes:
but the first number is equal to—1 and the second is > 0 if Dy Ф
The last assertion of (i) is obtained immediately by induction on r, starting
with r = 3. For r - 2 (in the notation of Proposition 25.1) Pic V is generated
by the classes Zo --h>h> an^ these are exceptional.
(ii). We keep the notation of Proposition 25.1. Suppose that/(D) is not a
point. If the class I of 0|<(D) is of the form then b^ (D, /.), so
that the point xi becomes a Z>z--multiple point on f(D) under the collapsing of
a curve of the class / into the point xz-. Because D =f~~1 we have
r
class/*(/(D)) = I + 2D bfa = alQ ,
z=l
from which we get for the degree of /(D) the expression
(/(D), Op2(l)) = (r(/(D)),Zo) = zz.
It is now evident that the list (ii)(a)—(g) is a direct translation of Table (IV.8)
in geometric language. Finally, Table (IV.9) is obtained from (IV.8) by a
straightforward calculation.
Remark 26.3. We shall say provisionally that the points Xp..., xr G P2
are in general position if their blowing up leads to a Del Pezzo surface. As was
proved in Theorem 24.5; r < 6 points are in general position if and only if no
three of them are on one line, and no six of them are on one conic.
Theorem 26.2 permits us to extend this list of necessary conditions for
‘general position’ until r = 8: the curves described in conditions (ii)(b)— (g)
must have exactly the described set of multiplicities in the points xz* (in the
‘non-general case’ these multiplicities may jump). I do not know whether this
list of conditions is sufficient.
An analysis similar to the one carried out in Theorem 24.5 can become
more complicated here because can cease to be very ample, and for a
projective embedding of Fby means of Qy” one must consider curves of
sixth and higher order.
Remark 26.4. The blowing up of r > 9 points on P2 can already lead to
surfaces V with infinitely many exceptional curves. For the exceptional
classes in Pic V it is not difficult to verify this, by examining the system of equa-
tions (IV.6). It is possible to show that on some surface V all classes are repre-
sented by curves, by means of the following geometric arguments (of T.R.
Safarevic):
We choose eight points on P2, leading to a Del Pezzo surface V of degree 8.
On it (Op1, Sip1) = 1, and the zeros of the sections of Sip1 form a bundle of
elliptic curves with one base point. Carrying out one more monoidal transfor-
mation with as centre this base point, we obtain a Neron model for the general
curve of this bundle. Its sections, and only these, are exceptional curves,
and the composition law of the general fibre induces on them the structure
of a free Abelian group of rank 8. This situation is studied in more detail
in Manin [2].
In the following chapter, the case r - 00 emerges in a somewhat different
context. It appears that the process of blowing up ‘all points of the plane’,‘all
points of the surface obtained’, and so on, admits a reasonable and important
algebraic interpretation, and it leads, for example, to a description of the
group of birational maps of a surface onto itself.
Returning to the main theme, we shall now study the group of automor-
phisms. We shall not recall the statement of Theorem 23.9 (i), which has al-
ready been proved.
26.5. Proof of Theorem 23.9 (ii). We shall provisionally denote the groups
described in conditions (ii)(a) — (c) of the theorem by Ifp respec-
tively.
There is an obvious embedding C and a homomorphism W1 W2
which maps each automorphism of {Nr,cor,( ,)} onto the corresponding
induced permutation of Ir. We shall show that both these maps are isomor-
phisms.
The kernel of W1 -> W2 is trivial because Ir generates (Theorem 26.2 (i)).
On the other hand, let Z/r be the free Abelian group generated by Ir, let
ZIr ->Nr be the natural epimorphism, and let Kr be its kernel. We have
К ~ aiei^ Ve^Ir , S- zz.(e., e) = 0}, e EI .
f * Jr t fill If
This follows from the fact that on a rational surface linear equivalence of
divisors is the same as numerical equivalence. Of course this can also be di-
rectly checked from the definition of Ir.
Further,
= ar Zz e (modXr), 7
e^Ij.
where —a"1 = 120,28,9,4,2,1 for r - 8, 7, 6, 5,4, 3, respectively. In fact, it
is clear from table (IV.8) that for r = 8, the map e - e, and for r = 7 the
map ен — 2cor-e sends Ir into itself (without fixed points). For r < 6, the
coefficient ar can be directly calculated without difficulty.
This description of Kr and shows that any permutation of the vectors
G I which preserves their pairwise intersection numbers induces an auto-
morphism ’LIr/Kr ^Nr. This automorphism preserves intersection numbers
and so thfct JVj -> is an isomorphism.
It remains to show that Wy contains no elements other than those of
K3 = JV(7?r). For r = 7,8, the full group of orthogonal maps of coj; which send
Rr into itself (‘the extended Weyl group’ JV*(Ar)) coincides with JV(Ar) (see
Serre [4]). However, for 3 < r < 6 the index of IE(Rr) in IV*(7?r) is equal to
2 because changing the signs of the roots, I —Z, is not generated by the re-
flections Sj. We show that it does not induce an admissible automorphism of
Nr. Otherwise this automorphism would be the reflection with respect to the
vector ccy in the space R ® Nr which sends the integral lattice Nr into itself.
Because the intersection number is a unimodular form, this is only possible
for (сог, ccy) - ± 1, ± 2, that is, precisely for r = 7,8. Theorem 23.9 is now com-
pletely proved.
26.6. We write down the orders of the group ^(7?^) from the tables in
Bourbaki [ 1 ] (for r - 2 it can be directly calculated):
r 8 7 6~ ~5 4 3 Г”
1И'(Лг)1 21435527 21o345-7 27345 273-5 233-5 223 2
A first result on transitivity is the following:
Corollary 26.7. (i) Under the natural inclusion lr_^ - IrC\ ©JZqZZ,- C 7r,
the group W(Rr_f) is identified with the isotropy subgroup of the vector
lr&IrinW(Rr).
(ii) IV(^r) acts transitively on lr.
Proof, (i). It is obvious that all reflections I € Ir_y, leave lr fixed. There-
fore IV(7?^) is contained in the isotropy subgroup. Conversely, any element
of the isotropy subgroup induces an automorphism-of;
sub-object of {Nr,cor( , )}and therefore belongs to according to
Theorem 23.9.
(ii). The order of the W(Rf) orbit of the vector lr is equal to the index
: IV(^-i)) because of part (i). Comparing these indices, calculated
from the table in Section 26.6, with table (IV.9) of the orders of the Ir, we im-
mediately obtain the desired result.
More generally, every ordered subset ..., C Ir such that mj)
- 0 for к £j will be called an exceptional subset of length L It is clear that
such a set generates a Euclidean subspace of dimension i in R Ф As the in-
tersection number is indefinite on R ® 7Vr, z < r. As before we assume that
3<r<8.
Corollary 26.8 (i) The group W(Rr) acts transitively on the collection of
exceptional subsets of length iifi^r—l. If this condition is fulfilled, every
exceptional subset is contained in one of length r.
(ii) The group W(Rf) has two orbits in the collection of exceptional sub-
sets of length r—1: one of these consists of all maximal sets of this length and
the other of all поп-maximal ones.
Proof. We use induction on z. As in Corollary 26.7,L {= L Л ®r~l^Zl CI,
J ? r—l r К-0 к ri
and h/(Ar_z) is included in IV(Rz) as the isotropy subgroup of the set
(^•-i+1* • • •» (r)*
Corollary 26.7 clearly leads to a proof of transitivity up to length r—2. The
step from r—2 to r — 1 is not possible on account of the fact that IV(Rr) does
not act transitively on 12, and has two orbits: see the graph of Г2 in Section
26.9. One orbit consists of two non-intersecting classes, the other of one class.
This suffices for assertion (ii).
The following is an algebraic—geometric explanation of the meaning of a
maximal exceptional set of length r— 1: collapsing the corresponding curves
on a Del Pezzo surface of degree 9—r, we obtain the quadric P1 X P1 and not
the plane P2.
These corollaries are usually used as follows. Suppose we want to find out
how to construct the set of lines on a cubic surface V which intersect some
pair of crossing lines. We can take any pair, e.g. and Z2, and read off every-
thing needed from Table (IV.8). In Section 30 weshall develop a whole series of
such arguments. There we shall also implicitly use some other assertions on transi-
tivity, for example, on pairs of intersecting lines, etc. We leave it to the reader
to formulate and prove such results.
26.9. Graphs of exceptional curves. We attach to the set Ir a graph I). the ver-
tices of which are in one-one correspondence with the vectors of /r, and the
number of simplices joining I with l' is equal to the scalar product (Z, Z').
It is sometimes useful to examine the following pictures for small r:
Here the vertices are labelled by the images of the exceptional curves on the
corresponding Del Pezzo surface under the map К -> P2: the хг- are the blown
up points, the Zzy the lines passing through Xj.
Unfortunately, it is already impossible to depict Г4 in such a way that its
various symmetries become evident. It is even more difficult to draw Г5:
4 4 8 8
•— • p . 3 3 *5’ ee 2 2 » - ---• 7 7 • ;—• 6 6
• • 1 i • • 5 5
Here the simplices which join the vertices of the column on the left with the
vertices of the column on the right are not shown. They are reconstructed thus:
The vertices of each column are connected with precisely one vertex of each
pair in the other column in the following way. The left (right) vertex of each
pair is joined with the left (right) vertex of the pair in the same row of the
other column and with the right (left) vertex of the other pairs. For example,
1 is joined with 5, 6,7,8 (in addition to 1).
After some trouble, the reader can check that precisely the same incidence
relations of exceptional curves on the image of a monoidal transformation of
P2 with centre , x5 are enumerated as follows:
1 2 3 4 5 6 7 8 12345678
^01 X2 *3 * X4 *1 Z02 ^03 ^04 *0 ^12 ^13 ^14 $ ^34 ^24
(q is the conic passing through x1?..., x5.)
The graph Г6 of the 27 lines (for us the most interesting graph) is al-
ready very complicated, and I cannot draw it. Still more complicated are
Г7 (56 vertices) and Г8 (240 vertices), where, in addition, vertices occur which
are joined twice (in Г7) or thrice (in Г8) by simplices. However, the involution
/ —2cor — I of the graph Г7 permits us to reduce it to a graph of almost the
same order as Г6: r7/Z2 consists of 28 vertices, and the simplices reflect the
incidence relations of the 28 double tangents to a smooth quadric on P2.
A simple explanation of this is the following: Project a cubic surface
V С P3 onto some plane P2 С P3 from a point x G V that does not lie on P2
or on one of the lines of V. Resolving the unique singularity of this map by
blowing up the point x, we obtain a morphism V' -+ V P2 of Del Pezzo sur-
faces of degree 2 on P2. The degree of this morphism is equal to 2, its
branching curve, the projection on P2 of the ‘apparent contour’ of V from x,
is a quartic. The exceptional curves of V1 are pairwise glued together and they
are projected onto a double tangent to this quartic: 27 of them are essentially
the images of pairs (a line on K, a conic on V passing through x and the line)
and the last one is the image of C(x). For some more detailed calculations see
the proof of Theorem 29.4.
Here is a reformulation in terms of the graphs Гг of some of the results
proved:
Let v G Гг+1 be any vertex of the graph. Then the subgraph of the vertices
which are not joined to v, together with the simplices joining them, is isomor-
phic to Гг
The group РЕ(АГ) is isomorphic with the full group of automorphisms of
the graph Гг. (The action of an automorphism on different simplices joining
one and the same pair of verticesTsmoTtaken into account; such simplices
only occur for r - 7,8.)
The group N is isomorphic to a quotient group of the zero-dimensional
chains of the graph Гг:
Nr = С°(Гг)/{2и.еГгu.l VK> Ъа.(»,, = 0},
where (vz-, uK) is the number of simplices joining иг- to uK,
27. The zeta function
Let & be a finite field of q elements and let V be a smooth projective sur-
face over k. We denote by F the canonical generator of the Galois group G ~
- Gal (к /к) : F(z) - zQ for all z G к. This is the Frobenius automorphism; it
acts on 7V(K) = Pic (7® k). We denote by N the number of k-points of the
surface V. ’
Theorem 27.1 (A. Weil). If the surface V ® к is birationally trivial, then
N = q2 + q TrF*+l ,
where Tr F* denotes the trace of F in the representation of Gai (k/k) on N(Vf
Corollary 27.1.1. (i) N s 1 (mod q). In particular, the set V(k) is non-
empty.
(ii) \N~ (q2 + 1)1 <rq, where r = rk7V(7).
Corollary 27.1.2. Let Na be the number of geometric points of V over the
field of qa elements, «=1,2,... . Then
(oo \
Z) Nafla\ = [(1-0(1-<f2r)det(l — qF*t)}~x ,
a=l f
•where the characteristic polynomial det (1— qF*f) refers to the representation
ofFonN(V).
Proof. Applying Theorem 27.1 to the surface V ®^ka (where [ka \k]-a)
over the finite field ka, we obtain:
Let/p ... , f G Cbe the characteristic roots of the representation of F* on
C 0 AT(F). As the group TV(K) is finitely generated, all its elements are defined
over some finite field k^. ThenF^ acts on jV(F) as the identity, so that all the
f are roots of unity. Assertion (ii) of Corollary 27.1.1 follows from this.
Further Tr F*a~ Therefore
Z/ - Z/ q^ata /a + 22 qfff tal a + Z/ ta\a
a~l /7=1 z-1 a~l
r -i
=-log (1-<?20(1-дП (i-?40 ,
Z=1
which proves what we want.
27.1.3. Proof of Theorem 27.1. We start with a survey of some necessary
algebraic—geometric results.
27.1.4. Intersection number. Let К be a smooth «-dimensional projective
variety over some field k. The free Abelian group generated by the irreduc-
ible subvarieties of Vof codimension к is denoted by CK(V). Its elements
are called cycles о/(pure) codimension к.
The subvarieties ZpZ2 of codimensions к, «-к, respectively, are said to
be in general position if their intersection is zero-dimensional and in the
local ring 0x of each point xEZ^ Л Z2 there exists a system of generators
of the maximal ideal (Д,..., fK ; , gn„K) such that the first к ele-
ments give a local equation for Zj, and the remaining ones give one for Z2.
Geometrically this means that the intersection is transversal. The sum
^xeZi ciZ2 (^(x) : *s ^en called the intersection number ofZp Z2.
The cycles Z^ G CK(F), Z2 G Cn~K(F) are in general position if each com-
ponent of Zj is in general position with respect to each component of Z2.
For such cycles the intersection number is extended by bilinearity.
The first step of the Weil theory consists in interpreting TV as an intersection
number. Let k be a finite field of q elements.
We denote by A C VX V the diagonal, and by A^ C VX V the graph of
the ‘geometric’ Frobenius morphism Ф : V -> V, which on an affine subscheme
Spec (Л) is induced by the endomorphism?! (note that it preserves
the ^-algebra structure of A).
Lemma 27.1.5. The cycles A, A^G СИ(КХ V) are in general position and
the number N of к-points of the variety V is equal to
7V = (A, A(1)) .
Proof. It is clear from the definition of Ф that
△ П A(1) = {(x, x) e V X 71Ф (x) = x} а У(к)
(as a set theoretic intersection). Let x be a closed point on V of the first degree,
0x its local ring on V, and Zp ..., tn the generators of the maximal ideal of
0x. We denote by ty: VX VVthe projection, and let t^ -Pj(tk),
i = 1,2. Then ... , t$P, ... , are generators of the maximal ideal
m(xxy in the ring 0^x xy on VX V. The local equation for A has the form
(/^1) ) and A^) is defined by the equations (t^Q ~ t^\
... itfflQ— 7^).It is clear that these equations are independent modni^. xy
which proves the desired result because
(△,△<») = Z) (k(xy.k) = N.
x^V(k)
27.1.6. The lemma just proved naturally leads to the idea of establishing
an analogue of the Lefshetz formula for the number of fixed points (here of
the endomorphism Ф) and to apply it to the calculations of TV. The corre-
sponding formalism, however, is essentially linked with the possibility of re-
placing A and A<I) with cycles of the form Sz-ZlzX Z2z, where ZKZ- C C*(K),
without changing their intersection number. In topology this can be done by
passing to cohomology classes and using the Kiinneth formula H*(VX V)
~/f*(V) 0 JT*(V). A. Grothendieck has proved that there exists a construc-
tion of cohomology groups with the necessary functorial properties also in
algebraic geometry over an arbitrary (in particular a finite) field.
In our particular case, however, we can get by without this profound theory
of Grothendieck, roughly speaking, because on rational surfaces and their
products ‘all cohomology classes are algebraic’. —---- ----- —
Before stating this result precisely, we shall first explain how algebraic
cycles are replaced by classes. A cycle Z E CK(V) is called (numerically) equiv-
alent to zero if (Z, Z') = 0 for all Z' G Cn~K(V) in general position with re-
spect to Z.
Lemma 27.1.7. The cycles equivalent to zero form a subgroup Zq(F) C
CZ*(7).
For every two cycles Zj GZK(K), Z2 G Zn~K(V), there exists a cycle
Zq EZq(F) such that Zу + Z$ and Z2 are in general position. The intersection
number (Zj + Zo, Z2) does not depend on the choice ofZ®.
Corollary 27.1.8. We set AK(F) ~ ZK(V)fZ^(V), Then the intersection
number induces an everywhere defined non-degenerate pairing
ЛК(7)Х An~K(F)^Z,
We omit the proof of Lemma 27.1.7.
Lemma 27.1.9. Let V be a birationally trivial surface over an algebraically
closed field k. Then
A°(V) = A2(V)^Z,
A1(V) = PicV.
Proof. The map D class (Op (D)) preserves intersection numbers of divi-
sors which are in general position. On the other hand, under our conditions
the group Pic V is free with a finite number of generators and the intersection
number is non-degenerate on it. The desired result follows easily from this.
Lemma 27.1.10. Let Kp K2 be two birationally trivial surfaces over an al-
gebraically closed field. Then the Kunneth formula holds:
Л"(Г1ХИ2)= ® А^®А’(У2), 0<h<4.
i+j~n
Here, Zy 0 z2 denotes the class represented by the subvariety Zj X Z2 if
the classes z^ eAl(V\), z2 GzF(F2) are represented by tne subvarieties
Z^ G Fj andZ2 G F2. ...........— —~-------------------------- .
I do not know a simple proof of this result. It can be deduced without
trouble from a general formula of A. Grothendieck which permits the cal-
culation of the Chow ring of a monoidal transformation of a given variety.
In fact, X F2 can be obtained from P2 X P2 by a chain of monoidal
transformations ‘along the fibres’. The result follows from this by induction
on rk Pic + rk Pic F2.
Another variant consists in using an algebraic ‘cellular decomposition’ of
X F2 and ‘excision axioms’ for the Chow ring A *(K), as in the expose of
Grothendieck [1], For a more detailed study, see Manin [1].
Using Lemmas 27.1.9 and 27.1.10, we now calculate the classes of A, A^)
and their intersection number.
Lemma 27.1.11. Let (zz-) be an orthogonal basis in the space C 0 Pic V,
V~ V®k, Le., (zp Zj) = {Kronecker symbol). Then
class A = ly 0 ey + zD z2- 0 zi + ey 0 1 v G C 0 А2 (ИХ И) ,
z=0
where ly, ey are the canonical generators of the groups A°(F) and А^(И):
the classes of V and of a point, respectively.
Proof. Let Zp Z2 GA^F) be cycles in general position. It is not compli-
cated to check, as in Lemma 27.1.5, that then the cycles A and Zy X Z2 are
in general position in FX Fand that (A, Zj X Z2) = (Zp Z2). In fact, that
А П (Zj X Z2) coincides with Z^ О Z2 is obvious, and passing to the local
ring at the point x G А П Zj X Z2 linearizes the problem. Consequently,
(class A, 1 у 0 ey) = (class A, ey 0 10 = 1 ,
(class A,zz 0 z.) = (z., z^ = .
But the element written down in the statement of the lemma has the same
intersection numbers with the basis (Ip 0 ey, ey 0 lp, z;- 0 Zj) of the group
C 0 A2(F). The desired result now follows from the fact that the intersection
number is non-de gene rate and that A2(F X F) has finite rank.
27.1.12. We now compare the two Frobenius automorphisms: the ‘algebraic’
one F: V and the ‘geometric’ one Ф : V -> K. Let U = Spec (Л) С V be an
open affine subset, and let U = U ® к С И Then these morphisms act as fol-
lows on the functions S a.g. G к 0, A, a. ek,g.€A :
I I К I I
^~l(^aigi)=Ya9gi,
ф-1 (Sa.g^Sa.g^1.
Therefore their product is the ‘complete’ Frobenius morphism which acts on
Oy as ‘raising to the power q\
We now remark that for any morphism of k-varieties : Pj -> K2 of the
same dimension, one can define two maps
ip* : AK(V2)-^-AK(Vl) ,
^-.A^V^A^)
with the usual functorial properties. Both these maps are even defined on
cycles. In particular, for a subvariety Z C we have ip.*(Z) ~ dip(Z), where
is the set-theoretical image, and d is the degree of the map Z -> <p(Z)
(zero if dim ^(Z) < dim Z). Finally, if all elements are defined, the following
‘projection formula’ holds for intersection numbers:
(/Zp z2) = (Zj, ^(Z2)),
which holds without restrictions for the cycle classes.
We apply this formalism to the morphism id^ X Ф : V X V -* V X И By
definition, - (id^ X Ф)* A. Therefore
N = (△, △(’)) = (△, (idKX Ф\Д) = ((id v X Ф)*Д, △) .
We transfer this identity to classes of cycles. Note that (idj/ X Ф)*(^1 X Z2)
= Z1 0 Further, it follows from the discussion on the action of F
and Ф that F* о Ф* : Pic V -> Pic V coincides with multiplication by q, so that
Ф*(?) = (z) for all z GPic V. Moreover, Ф*(10 = 1 у, Ф*(ер) = q2?y
(q2 is the degree of the morphism Ф). Combining all this, we find:
N = q2 + q E (F*)-^) + 1 = q2 + q Tr F* + 1 ,
because Tr(F*)"l ~ Tr F*: the representation has been obtained by an exten-
sion of constants from a real one (even an integer one). This proves Theorem
27.1.
27.2. We now deduce from Theorem 27.1 the corresponding global results.
Let |F: Q] < °° and let V be a smooth projective surface over к which is bira-
tionally trivial over k. We denote by К Э к a finite normal extension such that
all generators of the group Af(F) - Pic (V ® k) are already defined over K.
Let x be the character of the representation of the group Gal (K/k) on N(V\
Theorem 27.3. The Hasse-Weil zeta function of the surface V coincides up
to a finite number of Euler factors with the product
f 0Л) №-2, k) L (s-1, X, к/к) ,
where L is the Artin function corresponding to the character x.
Proof. Everything reduces to a careful comparison of the definitions.
(i) Artin *s L-function. Let А С к be the ring of integers, let pv C A run
through the prime ideals of A, and let be the order of the field A/p^ The
function L - ПуТу is the product of the local factors. They are easily described
for those v, corresponding to the ideals pv, which are unramified in K. Let В be
the integral closure of A in К and let С В be any of the prime ideals lying
over Py. The automorphism z >->z^u of the extension B/Pv over A/pv is in-
duced by some element Fv G Gal (K/k) = G, which is defined up to conjugacy.
We fix some finite-dimensional linear representation g >-> g* , G -> Aut M,
overC. The value of det (1—F* t) does not depend on the arbitrariness of the
choice of Let x be the character of the representation. Then, by definition,
Lv{s, x, K/k) = det (1 - (IV.10)
The finite number of factors which correspond to the ramified and the Ar-
chimedean v are more difficult to define; here there is no necessity to go into
the details. Instead of К we can take k, inserting the obvious modifications.
(ii) The Hasse- Weil function. There exists a model of the surface V ‘over
A\ that is, апЛ-scheme Fq of finite type, projective over Spec A, such that the
general fibre’ V$ ® A Spec к is isomorphic to V, For almost all pv C A, ‘the
reduction’ ® a sm0°th surface over A/pv = k(v) which is
birationally trivial over A/pv = k(v). Even more, for different choices of
models Vq their reductions are isomorphic for almost all v- we shall simply
denote them by Vv. The local factors of the Hasse—Weil zeta function are by
definition equal to
zrXS) = (1-O_1(1 det(l-^F*) (IV.ll)
(cf. Corollary 27.1.2). Here F*is the representation of the local Frobenius
automorphism on 7V(F ).
Comparing formulae (IV. 10) and (IV.l 1) shows that to identify the local fac-
tors it is sufficient to construct a commutative diagram
Pic (V ® k) Pic (И ® к(у))
F* F*
V V
Pic (V ® k) Pic (K ® k(v)),
(IV.l 2)
in which the map is an isomorphism for almost all v.
We choose a finite number of divisors С V ® к such that they are pair-
wise in general position, and such that the classes of the 0(P^ generate
Pic (V ® k). As they are given by a finite number of equations over a finite
open covering of V, it is not difficult to check that for almost all v one can de-
fine their reductions: the divisors Dz y C Vv, while the intersection numbers
(Dif Dy) coincide with (Dz* v, D? v). Mapping the class of 0^(Dy) into the class
Oy (Dz- y), we obtain a homomorphism This is clearly an inclusion. A com-
parison of the actions of G on V ® к and of F* on К ® к (и) shows that the
diagram (IV. 12) is commutative because these actions are already defined on
divisors.
It remains to show that the <pv are isomorphisms. First of all the ranks of
the groups Pic (K® k) and Pic (Kv ® к(рУ) coincide: for instance, because
the first one is equal to 10— (Пр, П^), and £2^ reduces to for almost
all v. This is already sufficient for comparing (IV. 10) and (IV.l 1) because
it follows from this that C ® <pv are isomorphisms. The triviality of the co-
kernel of however, also follows easily from the fact that the ‘intersection
number’ form has determinant ±1 already on the image of This proves the
theorem.
28. Minimality and classes of conjugate elements in Weyl groups
Let К be a smooth projective surface over a field k. Obviously rk Pic 1;
the class of Oy (1) for any projective embedding has infinite order. If rk Pic V ~
= 1, then the surface Vis minimal: nothing on it can be collapsed. B. Segre
discovered that for cubic surfaces the opposite result is also true:
Theorem 28.1. Let V be a cubic surface over a perfect field k, and let
G = Gal (k/k\ The following assertions are equivalent'.
(i)7V(7)G =ZwK.
(ii) V is minimal.
Proof. It is clear that (ii) follows from (i). To prove the opposite implica-
tion, we suppose that N(V)G Because (coy, co у) = 3, co у is not
divisible inTV'(U); consequently, there exists an element n EN(V) which is
linearly independent of coy.
Let I EN(V) run through all exceptional classes. Then the intersection
number (n, I) takes at least two different values; otherwise from the equality
(n + (n, Z) coy, I) = 0 for all I it would follow that n - — (n, l).coу, contrary to
our assumption.
We set
a = min (n, I) ,
I
b = max (n, I) ,
I
D.= S I.
(n,iy=b
Each of the classes Da, Db is represented by a unique positive divisor on V
the cdmponents of which are lines L with {n, I) = a or b (because the divisors
are G-invariant; cf. Lemma 21.8.1). For brevity’s sake we shall denote these
divisors also by Da, Db and we shall show that one of them can be collapsed
over k.
Suppose this is not so. Then there exist two different exceptional classes
la, l^ (components ofL^) such that (la, l'a) = 1 (Theorem 21.8; the intersection
number of exceptional curves on V cannot be greater than one). Analogously,
we can find components lb and lb of Db with (Z$, lb) = 1. We set
ZQ = - coy - la - la , 1^ coy - lb - lb .
One immediately verifies that these are exceptional classes (their geometric in-
terpretation : Zq is the class of the third component of the intersection of V
with the plane passing through the lines which represent la and Va ).
From this we obtain
(и, - wr) = (и, la + l'a + Zo) = 2a + (и, zo) < 2a + b,
(n, — Шу) = (n, lb + l'b + Zj) = 2b + (n, Zp > 2b + a,
which contradicts the inequality a < b. The theorem is proved.
Remark 28.1.1. This result is not true for minimal Del Pezzo surfaces of
degree d = 1,2,4: the rank of the group _V(F)g can also be two.
28.2. We now use some arguments, connected with minimality, to introduce
a convenient principle for the classification of conjugate elements in the Weyl
group Tables for the classes for r = 6,7 can be found in the works of
Swinnerton-Dyer [2] and Frame [1]. Cartier [1] contains tables for all Weyl
groups. As always, when there is question of a finite number of data, the meth-
ods of organizing them into a table are in principle indistinguishable from each
other. The principle proposed here firstly gives the possibility to use induc-
tion on r and secondly is convenient for the calculation of the algebraic—geo-
metric invariants described in Section 23.11. A table for FV(A6) will be given
in Section 31; it would be worthwhile to compile the analogous tables for
IP(R7) and W(R8).
We now turn to the precise definition. We shall consider the system
/if c Rr+1 = R <8>
Nr as given by its geometric realization, as described
in Sections 25 and 26; let Ir CRr+1 be the set of exceptional vectors. We con-
sider some element 5 G W(Rr). A set ..., mK) &Ir is called s-exceptional
if:
(i) (my ... , is s-invariant;
(ii) mj) - 0 for i Ф j.
(compare the definition preceding Corollary 26.8). The index i(s) is the maxi-
mal order of an s-exceptional set.
Suppose that s is an element of the group Gal (k/k) acting on Pic (F ® к ).
Let К D к be the field of invariants of the cyclic subgroup (sw). Then i(s) is
the maximal number of geometric components of a divisor on V ® К which
can be collapsed over k. In particular, if i(s) = 0, the surface V ® К is minimal.
For ф) > 0 the corresponding divisor can be collapsed by a morphism
VV9, and Pic К® к is obtained from Pic V' ® к in the well-known man-
ner.
These considerations are given concrete form in the following theorem:
Theorem 28.3. Let r > 3, and let Z$, Zp . .. , Zr be the standard basis for the
space Rr+1. Choose a number 0 < i < r and let denote the group of permu-
tations of the vectors ..., Zr). Let be embedded in W(Rrf
as described in Corollary 26.7, as the isotropy subgroup of (Zr__z+p..., Iff
(i) For every pair s C W(Rr_i), t G Sj, there exists a unique element in
WiRf) which acts as s onl§,..., lr_it and as t on lr^yf. .. , lr.
(ii) The set of elements of the type described forms a representation of
all classes of conjugate elements of index > i in W (Rr), provided that i^r~l.
Proof, (i) (a) Existence. W(Rr_j) C JV(Ar) coincides for all i with the iso-
tropy subgroup of the set (Zr_/+1,... , Zf); on the other hand, is included
in И^/^) as the subgroup of all permutations of this set which act trivially on
Zo,... , lr~i (proved as in Section 26.5). The product st, s <= tesh
under such an inclusion therefore possesses the desired property.
(b) Uniqueness. An element of IP(Ar) acting trivially on'Z0,... , lr_i9
lr-i+1’ • • • , Zr is the indentity on Rr+1.
(ii). Let 5 G W(/Q be an element of index z, and let (mp ..., mf) C Ir be
a maximal ^-exceptional set. According to Corollary 26.8, for i ¥= r - -l it is
possible to extend it to an exceptional set of length r (not necessarily 5-in-
variant). Sending this maximal set by means of w E JV(7?r) into the part
(Zp ..., lr) of the standard basis of Nr such that (flip..,, mz) go into
(Zr„z+p ..., Zr), respectively, we obtain that the element maps the
set (Zr_/+p..., Zr) into itself. Therefore it has the form described in (i).
The theorem is proved.
28.4. Theorem 28.3 permits us to give the following recipe to describe the
classes of conjugate elements in ИД7?Г):
(i) . Write down the classes of elements of index 0 in hZ(Rr_j-) for all i < r.
(ii) . Draw up the list of pairs (class of elements of index 0 in
class of permutations on i letters).
(iii) . Write down the classes of elements of index r—1 in JV(/?Z).
(iv) . Make explicit which elements of (ii), (iii) are conjugate with one an-
other in IV(ftr).
The result of this work for If(Ag) is contained in Section 31. For r = 7
or 8, one essentially only needs to write down the classes of index 0; the in-
formation on ^(A^) can according to Theorem 28.3 easily be recast as in-
formation on the classes in W(Rf) and IV(Ag) of index > 1 and > 2, respec- .
tively.
We formulate as a theorem some properties of the classes in Ж(/?6) which
I detected by looking closely at the table.
Theorem 28.5. (i) The index of an element s € Pf(A^) can only take the
values 0, 1,2, 3, 6.
(ii) The elements s, t G IV(A 6) are conjugate if and only if their sets of
characteristic roots in the space C ® coincide.
♦
Proof. See Table 1 in Section 31. The first result could also be established
by direct arguments, which are left to the reader as an exercise. I do not know
how to prove the second fact without sorting out all classes: it is possible
that it is also true for r - 7,8 or even for all Weyl groups. For instance, for
a symmetric group it is obvious since the set of characteristic roots can be
directly reconstructed from the set of the lengths of the cycles which occur in
the decomposition of the permutations of a class.
29. A cohomological invariant and the degree of unirationality
Let К be a smooth projective surface over a field k,G = G^\(k/kj. In this
section we shall show that the group Z/^G, 7V(K)) is a useful birational invariant
of V and shall in particular verify Theorem 23.3. An interesting interpretation
of this group and its further applications will be given in Chapter VI.
Theorem 29.1. Let f\ be a birational morphism of smooth projec-
tive surfaces over k. Then the map of G-modules f* : N(T) induces
an isomorphism of Galois cohomology groups
f: /fi(G, N(V’)) (G, N(V)) .
Corollary 29.1.1. The maps fV(G, N(V)) extend to a (contravariant)
functor on the category of smooth projective k-surfaces with k-birational maps
as morphisms. This functor takes every birational map into an isomorphism
of the groups .
[To define : Я1 for a birational map : V’ -> V one must con-
sider some resolution of it,
and set = g*o (/z*)~l, using the fact that Л* is an isomorphism. The
independence of the choice of the resolution is checked by standard diagram
chasing.]
Theorem 29.1 is a rather essential weakening of the following observation.
Let H G G be some open subgroup of finite index. The Я-module Z (with triv-
ial action) then induces a G-module Z [G] ® ^[Я] ran^ : H] on a
basis of which the group G acts by permutations, as on G/H. A direct sum of
a finite number of such G-modules shall be called trivial.
Lemma 29.1.2. Under the conditions of Theorem 29.1 there exists a trivial
G-module M such that
N(V’)^N(V) ®M.
Vxwt.f* embeds 7V(KZ) in 7V(K) as a direct summand; its complement
7И= Ker Д is generated by the classes of7V(K) which are collapsed (under /)
and which are independent over Z; there are a finite number of them, and they
are permuted by the action of G on them. This proves the lemma.
29.1.3. Proof of Theorem 29.1. According to Shapiro’s lemma on the coho-
mology of induced modules
Я1 (G, Z [G] ®z [Я, Z) =s Я1 (Я, Z) = Hom (H, Z) = 0 .
Therefore (G, M) = 0 for every trivial G-module M, so that Theorem 29.1
follows from Lemma 29.1.2.
Corollary 29.1.4. If the surface V is birationally trivial, then N(Vf)=
= 0 for all closed subgroups HCG.
(In fact,#1^,^2)) Z) = 0.)
Using the tables of Section 31 and this criterion, it is easy to construct
birationally non-trivial Del Pezzo surfaces, in particular cubic surfaces. We
now show that ZZ1 can also give more precise information on the birational
properties of V,
Theorem 29.2. Let Vbe a smooth projective surface and let there exist a
rational map of finite degree \p: P2 V. Then the degree of<p is divisible by
the least common multiple d of the exponents of the groups Я1 (H, N(V))for
all possible closed subgroups HCG.
Proof. We remark first of all that the surface V ® к is birationally trivial
over к, so that 2V(F) is a free group with a finite number of generators which
are all defined over a finite normal extension К Э к. Therefore all the groups
Я1 (Я, 7V(K)) are finite. Further, let be the subgroup corresponding to K.
Then it suffices to take for H a subgroup of the form С H C G and then
Я1 (H, ~ Я1 (Я/Gq, Я(Г)) •
(The standard ‘restriction—inflation’ sequence; see "‘Algebraic Numbers', Ch.IV.)
The calculation ofd is therefore a completely finite problem, and J is finite.
We now proceed to the proof. Let
be a resolution of singularities of the map where g is a birational morphism
and A is a morphism of degree deg This diagram defines two homomorphisms
of G-modules:
A* :JV(K) МП,
M : МП 7V(7) ,
and it is clear from the definitions that h*h* is multiplication by deg A ~
= deg <p in the group N{V). But Я1 (G, МП) = ° (Corollary 29.1.4). There-
fore A*A* induces the zero homomorphism of the group H^(G, N(V))9 so that
deg annihilates this group. Consequently, deg у is divisible by the exponent
of H^(G, N(Vf). Applying this result to the map
:P20 K-+V&K, K = kH
for an arbitrary closed subgroup H C G, we obtain the desired result.
The proof of Theorem 29.2 shows that the following invariant of the sys-
tem Rr is important. Let Nr be the lattice generated by the vectors
. ,lr in the standard realization of Rr (cf. Section 25). This group is a
MM)‘m°dule. Let dr denote the least common multiple of the exponents of
the groups Я1 (G, 7Vr) for all possible subgroups G С IV(M)•
Theorem 29.3. The numbers dr are given by the following table'.
r <4 5 6 7 8
dr 1 2 6 2a .-3 , , (1V.13) 2* 3C 5d
where a, b, c>l,c?>0.
(The author has been too lazy to calculate the constants a, b, c, dl)
Proof (outline). Let p be a prime number. To see the contribution of p to
dr, it suffices to restrict oneself to the calculation of H\H, Nr) for p-sub-
groups/ZC MM)-
Some details of the calculations for r < 4 are contained in the following
section. The results of the calculations for r = 5 are given in Table 3 in Section
31; in particular, Я1 = 0 for 3-subgroups and 5-subgroups.
When passing from IV(T?5) to PE(7?6), the Sylow 5-subgroups and 2-sub-
groups do not increase (see the table in Section 26.6). Therefore d = 2 * 3K.
The table shows that к > 1. A more detailed computation, which is omitted
here, gives к = 1. When passing from IV(7?6) to JV(T?7), the 3-subgroups and
5-subgroups remain as before. The 2-subgroup increases, which can lead to an
increase in the power of 2 in dj. (I do not know whether this happens).
Moreover, there appears a 7-subgroup Gj (see the table in Section 26.6).
However, it yields no contribution to d^ for the following reason. It is cy-
clic of order 7 and therefore conjugate with the group of cyclic permuta-
tions of /j,... , Z7. The algebraic—geometric model of the situation shows
immediately that the G7-module 7V$ is isomorphic to ZZq ф Z [G7], so that
W7,tv8) = o.
Finally, the assertion concerning И7(7?8) follows from what has been shown.
(The 7-subgroup again does not increase.)
Theorem 29.4. Let V be a Del Pezzo surface of degree 9 — r over a field k
and for r > 5 suppose that there is a point in V(k) which is not on any excep-
tional curve. Then there exists a rational map tp:P2 V of degree br which
is given by the following table:
r | <4 5 6 7
Sr 1 2 6 24
(IV. 14)
(For the calculation of deg the case of small characteristic of the field k
has been omitted; in particular, for r = 6 the characteristic is taken to be dif-
ferent from 2.)
Remark 29.4,1. Comparing this with Theorem 29.2 and 29.3 shows that
for r < 6 the maps ip described above possess ‘generally speaking’ the lowest
possible degree. The stipulation relates to the circumstance that we have not
shown here that it is possible to realize all subgroups of 1^(7?^) as an image of
the Galois group in Aut 7V(K). Besides, examples which realize the largest pos-
sible exponent of H^(G,N(V)) can be constructed without difficulty by means
of Table 1 in Section 31, at least for r < 6. See also the discussion in Section
23.13.
Remark 29.4.2. The requirement that there exists a /г-point outside the ex-
ceptional curves for r ~ 5,6 will be removed in the next section.
Remark 29.4.3. I do not know whether there exists a unirational Del
Pezzo surface V of degree 1 for which the set /(&) is everywhere dense. Of
course, such a V must be minimal.
29.4.4. Proof of Theorem 29.4. We shall treat the cases for different r con-
secutively.
(i) r ~ 0. According to Theorem 24.3, V® к is isomorphic to p|. Such
surfaces are called Severi— Brauer surfaces; it is well known that the existence
of an isomorphism V ~~ P2 is equivalent to: V(k) is non-empty (see, for exam-
ple, Roquette [1]).
(ii) r = 1. According to Theorem 24.3, V ® к is isomorphic either to a
4 2 11
monoidal transformation of P^ with a closed point as centre, or to P^ X P^.
In the first case there exists a unique exceptional curve on V which can be
collapsed over k. This gives a ^-morphism V -> V\ where V' is a Severi—Brauer
surface; it is trivial if /(£) is non-empty, but this is always the case because
there is a Appoint on the exceptional curve. In the second case, after a mono-
idal transformation V* -> V with as centre a Appoint of the surface V there
appear on V’ ® к three exceptional curves with the configuration «—*—>.
The union of the two ‘extreme’ curves can be collapsed over k, which yields
a morphism Vr V" onto a Severi—Brauer surface, which is trivial, as above.
(iii) r = 2. The same argument as in case (ii). We remark that the excep-
tional curve ‘in the middle’ is isomorphic to P1: it is a form of P1 over к on
which there is an invertible sheaf of degree 1. Therefore for r - 2 the set
V(k) is always non-empty.
(iv) r = 3. If V is not minimal, everything goes as in one of the cases al-
ready treated. Let /be minimal. Considering the graph Г3 insection 26.9,
one easily convinces oneself that all exceptional classes are conjugate over k;
therefore there is no &-point on the exceptional curves.
Take a Appoint x G /(A;) and perform the monoidal transformation
V' -> V with centre x. It is evident from Corollary 24.5.2 that Vf is a Del Pezz6
surface of degree 5. An argument as in Theorem 26.1 or a look at the graph
Г4 shows that the curve D, the inverse image of x, intersects the exceptional
curves Z)1# D3 on к in precisely three points. These curves do not
intersect each other pairwise and the image of each of them on V ® к inter-
sects the pair of opposite sides of the exceptional hexagon Г3. Therefore the
divisor Dy +D2 + D3 is G-invariant and can be collapsed over k. This yields a
A;-morphism V' -> V", where V" is a Del Pezzo surface of degree 8, which re-
duces everything to the case r = 1, which has already been analyzed.
(v) r ~ 4. If there is a fc-point on V lying on the intersection of two excep-
tional curves Dp D2 on 7® k, then Dp D2 form a G-invariant pair. The set
of exceptional curves which do not intersect Dj orD2 on К® к is also G-in-
variant. A look at the vertices of Г4 in Section 26.9 shows that this set con-
sists of pairs of non-intersecting curves. Collapsing them over к reduces the
matter to the case r-3, which has already been treated.
Suppose now that x G K(fc) does not lie on any exceptional curve. Take the
monoidal transformation V* -> Vwith centre x; let Dbe the inverse image of x.
On V' ® k, the curve D intersects five pairwise non-intersecting exceptional
curves Dp ..., D5. (For instance, if under the collapsing И'® к ~>P~
D transforms to x0, and Xp..., x4 are the remaining centre points, then
Dp ..., D5 reduce to the four lines through x0, хг- and the conic through
х0, Xp ..., x4.) Collapsing the G-invariant divisor Dy + ... + D5 over к re-
duces everything to the case r = 0.
(vi) r ~ 5. Let x G K(&) be a point not lying on any exceptional curve. Then
the monoidal transformation V' V yields a Del Pezzo surface of degree 3,
that is, a cubic surface on which the line D, the inverse image of x, is defined
over k. The planes passing through D cut out a bundle of conics on V comple-
mentary to D. These conics are the fibres of a fc-morphism V' -> P1 (the base
of the bundle). The curve D is a section of multiplicity 2 of this bundle. The
surface Vf Xpl D is birationally trivial because there is a bundle of curves of
genus 0 on them, with one of the components ofDXplDasa section. The
projection V1XplD->K'is a morphism of degree 2.
(vii) r = 6 or 7.
(a) The construction of Let x G K(fc) be a point outside the exceptional
curves. Then there is a unique curve Dx of genus.0 over к passing through x
such that its inverse image under the monoidal transformation Vr V with
centre x belongs to the class — 2coy> -1 for r = 7 or the class —co у» -1 for
r = 6, where / is the class of the inverse image of x. (For r = 6, this is the inter-
section of V with the tangent plane at the point x).
Let у be the general point of the curve Dx; we consider it as a closed point
of the first degree on the surface F®*. fc(y) over the field к(7), and we denote
by Dy the curve on this surface constructed from у just asDx fromx. Finally,
let z be the general point of the curve Dy; we can consider it as a geometric
point of the surface V over a pure extension of к of transcendence degree 2.
Its locus is the general point of V, This yields the map <p.
(b) Calculation of the degree of y for r = 6. The degree of is the number
of inverse images of a general point. Choose in the containing space P3 D V a
coordinate system such that the point (1,0,0,0) is on К but not on one of the
lines of V, Denote by В the geometric locus of the tangent points with V of
the lines that pass through x and are tangent to V, generally speaking, outside
x. (Classically В is called the apparent contour of the surface V from the point
x.) We shall show that the degree of tp coincides with the degree of B, Indeed,
V contains the section Dx of the tangent hyperplane at some point x by means
of which <p is constructed. We construct tangent planes through the points of
Dx. The number of these planes which pass through (1,0,0,0) is also the num-
ber of inverse images of (1,0,0,0) under the map <p.
We now show that the degree of В is equal to six. The equation of И has
the form
TqI(T)+ T0q(T)+c(T) = O,
where /, q, c are respectively linear, quadratic and cubic forms in (T\, T2, Tf).
Any line through (1, 0,0,0) is uniquely determined by the coordinates of its in-
tersection point (тр T2, т3) with the plane To - 0. Let Го - r0, 7} ~ t±, i = 1,
2,3, be the parameter equation of this line. To find its points of intersection
with V outside (1,0, 0, 0), we get the equation
+ + rieW = 0 > T = (T1 > T2> T3 ) •
The tangency condition means that
42(t) - 4Z(t) c(f) = 0 . (IV.I 5)
[The simultaneous equalities c - q ~ I ~ 0 are not possible; otherwise there
would be a whole line on V passing through (1,0,0, 0).] We shall assume that
the characteristic of к is different from 2. Then the coordinates of the tangent
point look like
T0=c(t), _Tf =, z= 1,2,3. (IV.16)
The formulae (IV.16) define a morphism of the plane curve (IV.I 5) into
the space P3;2? is the image of this morphism. Those points of the plane 7q=0
for which c(f) = q(r) = 0 remain fixed under this map, and those for which
Z(r) = q(t) = 0 are mapped onto (1,0,0,0). Therefore the degree of B, being equal
to the number of its points of intersection with a general hyperplane = 0,
coincides with the number of solutions of the system consisting of eq. (IV.15),
the equation
3
aQc(T) - 4<7(t) Zz аг-тг- = 0 , (IV.17)
and the inequalities q(f) Ф 0, с(т) Ф 0. This number of solutions is clearly
equal to six.
(c) Calculation of the degree oftpfor r-l. This is done analogously. We
refer the reader interested in the details to the note of Sermenev [1].
30. Rational points
We start with a useful addition to Theorem 29.4.
Theorem 30.1. Let V be a Del Pezzo surface of degree 3 or 4 over a perfect
field k. If the set V(k) is non-empty, then it contains a point which does not
lie on an exceptional curve, provided one of the following conditions is ful-
filled'.
(i) The field к is infinite.
(ii) The field к is finite and has > 22 elements in the case of degree 4 and
> 34 elements in the case of degree 3.
Remark 30.1.1. The lower bounds for the order of the field are probably
not the best possible. Yet some bound is necessary, as is shown by the following
examples:
(i) . Let к be the field of two elements; there are in P2(fc) four points in
general position: (1,0,0), (0,1,0), (0,0,1), (1,1, 1). The remaining three points
are on lines through pairs of these points. Carrying out a monoidal transforma-
tion with the four indicated points as centre, we obtain a Del Pezzo surface of
degree 4 for which all points are on exceptional curves. This surface is, of course,
not minimal and birationally trivial.
Here is a more interesting example:
(ii) . Let к be the field of four elements, 9 € к, 9 Ф 0,1. The surface V is
given by the equation
7’3 + 7\3 + 7’3 + 9T% = Q .
U 1 Z J
All its ^-points are on the lines T/+ Tj* - 0, TK - - 0, where (f, jt к) is a
permutation of (0,1,2). Moreover, V is minimal (cf. Example 21.9) and bira-
tionally non-trivial, for instance because
H1 (Gal (k/k), Pic (F ® Z. X Z3 .
It is unirational over k? The construction of Theorejn 29.4 does not apply here,
and I do not know the answer.
30.1.2. Proof of Theorem 30.1. First let the field к be finite with q ele-
ments. Then the number of ^-points on V is not less than — aq + 1, where
a - 6 or 7 for degree 4 or 3, respectively (Corollary 27.1.1). On the other hand,
there are not more than 16 (# + 1) and 27 (q +1) points, respectively, on the ex-
ceptional curves. Comparing these estimates yields the desired result.
From now on we can therefore assume that the field к is infinite. We ana-
lyze the two cases each in turn.
Degree 4. If x G V(k) lies on precisely one exceptional curve, then this curve
is defined over к We collapse it and apply Theorem 29.4. Because a point of
V(k) cannot lie on the intersection of more than two exceptional curves, it
remains to analyze this case. Let x &Dy HZ>2, DjC V® k. The divisor
Dj +Z>2 is G-invariant. On the other hand, it is easy to see that there exists
exactly one more pair of exceptional curves D^, D^ on F® к such that
(Z>3, Z)4) = (Z>3, Dy) = (Z>4, Z>2) = 1. It is therefore also G-invariant;it satisfies
the relation 0^(7)^ + Z>2 +D3 + £>4) = Hp1.
Let D be a divisor on V such that D 0 к = Z>3 + D^. We have
(D,D) = 0, Pa(D) = Q, (Hr,L>) = -2;
moreover, dim//°(F, 0p(—D) 0 £20 - 0 because D is effective. The Rie-
mann—Roch theorem then gives
dim Я°(К Оу(РУ) > |(Д Z>) - D) + 1 = 2 .
Therefore the linear system of zeros of the sheaf 0y(D) covers all of the sur-
face K. Let D1 be an element of this system which contains the point x. Be-
cause (Sip1, Df) - 2, this is a curve of the second degree in the space which con-
tains И There is a simple Appoint x on it (for instance, because (Dp D') =
~ (Dp D) = 1). Consequently the setD^A:) is dense inD\ which proves what
we wanted.
Degree 3. Ifx G К (A) lies on precisely one line, then it is defined over A:;
by collapsing it, we reduce the problem to the preceding case. If л lies on pre-
cisely two curves, then the complement in the plane section through them is
a line defined over fc; collapsing it, we again obtain the desired result.
It remains to analyze the case where x lies on the intersection of three
lines on V ® k. (These points are called Eckhardt points: all Appoints of the
example 30.1.1 (ii) are of this type.)
Because the field к is infinite, in the containing space P3 Z V there exists
a line L through x which is defined over к and which intersects V at two dif-
ferent geometric points у, у which do not lie on the lines of V ® k. If these
points belong to K(A;), everything is proved.
In the opposite case, they belong to F(K), where K/k is a separable qua-
dratic extension, and are conjugate over k. The surface ECS^A'is unira-
tional over К in virtue of Theorem 29.4, so that the set V(K) is dense.
For every point z G V(K) we denote by z its conjugate point over k. We shall
show that it is possible to find points z, z in general position and such that
z о z ±x, (We cannot use the proof of 15.2: there we also supposed that
F(A) was dense.) Then, because о у = x, the closure of the set of points of
the form z о z is at least one-dimensional, because it is irreducible; on the other
hand, this set is contained in F(A:). This proves the desired result.
For the construction of z we denote by C(y) and C(j) the intersections of
F®£ к with the tangent planes at the pointsy, J, respectively. Further, we pass
through the line L ®кК'ъ plane P over К (in P^) such that P #= P and
P Cl C(y )-z Uy, z ¥=y. Then obviously P Cl C(J) = z U у and P Cl P =
= L Therefore the line through the points z, z G V(K) does not inter-
sect L ®kK, so that z о z =#x. This concludes the proof.
30.2. We now pass to the problem of the existence of Appoints.
Let [k : Q] <°°, i.e., к is a number field. A necessary condition for V(k)
to be non-empty is that all the F(A:U) are non-empty, where the kv are the
completions of A: for all possible valuations u. If this condition is also sufficient
for all varieties of some class C, then we say that the Hasse principle holds
for C-
In Chapter VI it will be proved that the Hasse principle clearly does not
hold for the class of minimal cubic surfaces. For non-minimal surfaces, how-
ever, we can say something positive.
Let К be a Del Pezzo surface over the field k. By analogy with the def-
inition in Section 28.2, the index of the surface V is the maximal number of
exceptional curves on V ® к forming a G-invariant divisor which can be col-
lapsed over k.
Theorem 30.3. Let V be a cubic surface,
(i) If the index of V is equal to 1, 2, 4 or 5, then the set V(k) is non-
empty and in the case where the index >2, V is birationally trivial
(ii) For the class of cubic surfaces of index 3 or 6 over a number field k,
the Hasse principle holds. The existence of к-points on such a surface is
equivalent to its birational triviality.
Proof. If the index of V is equal to 1, then there is a line on V together
with its ^-points. If the index is equal to 2, then the only case which is not
completely trivial is when the collapsible lines D +D are defined and conjugate
over a quadratic extension K/k of the base field. But let x € D (kf then
xED(K) and x о x E V(k). Theorem 29.4 then shows that V is biratio-
nally trivial because after collapsing D + D we obtain a Del Pezzo surface of
degree 5 with a point.
In the case of index 4, let be the collapsible four lines. Table (IV.8)
shows that there are precisely two lines which intersect all the Д. They do not
intersect each other and form a G-invariant set; we can therefore apply the
same arguments as for index 2.
Finally, let the index be 5, and let be the collapsible divisor. If as
the result of collapsing a form P1 X P1 is obtained, then Table (IV.8) shows
that there is a unique line intersecting all the D^, so that V(k) is nonempty;
on the other hand, after collapsing we can again apply Theorem
29.4. In the opposite case, there is precisely one line which does not intersect
all Dh and everything is obvious.
We now pass to assertion (ii). By collapsing an exceptional six lines on V
(in the case of index 6) we obtain some fc-form of the projective plane, that
is, a Severi—Brauer surface. It is known that the Hasse principle holds for these
surfaces. [Using this argument it is not difficult also to construct a surface of
index 6 without к-points: one must take a non-trivial Severi—Brauer surface
and perform a monoidal transformation with as centre the union of 2 closed
points of degree 3. It is known that, let us say, over a number field, such sur-
faces and such points on them exist.]
It remains for us to deal with the case of index 3. Collapsing the exceptional
triple of lines over k, we obtain a Del Pezzo surface of degree 6. Therefore it
suffices to prove that the Hasse principle holds for such surfaces. We shall
establish more precise results:
Theorem 30.3.1. Let V be a Del Pezzo surface of degree 6 over a perfect field
k. Let D С V be the divisor such that D® к is the sum of all the exceptional
curves on V® k. Then the open subset V \ D is isomorphic with a principal
homogeneous space over some two-dimensional k-torus.
Theorem 30.3.2 (Ono and Voskresenskii). The Hasse principle holds for the
class of principal homogeneous spaces over two-dimensional k-tori if к is a
number field,
30.3.3. Survey of the theory of tori. Let G = Gal(&/£), and let S be a left
G-module which is free over Z and of finite rank n. Its group ring over к is
isomorphic to к [S'] = к [Гр Tf"1,... , Tn, T~^ ]. The group G acts from the
left on к [5] — at the same time on the coefficients and the monoidals of
5, Let Л(5) - к [5]G be the subring of G-invariant elements. It is a fc-algebra
of finite type, so that T = Spec A(S) is an affine variety over к. T is called a
torus, and S' its group of characters. The following list of properties of T
justifies in particular its definition. The proofs can be found in the papers of
Ono [1], [2].
(i) . The canonical map к ®^A(S) -> к [S'] is an isomorphism. In particular,
T® Spec£[S] -G^ ,
whereGm ~ Spec k[T, T-1 ] is the multiplicative group scheme.
On T there is a unique group scheme structure such that T ® к ~
= Spec к [S] is an isomorphism of groups.
(ii) . T(k) ~ Hom (S', &*) as a G-module. Here, the action of G on T(k) is
defined such that
f(sx) = s(/(x)) for all fEA(S),x G T(kf s EG ,
and the action of G on Hom (5, к * ) is given by the usual formula
f(sx) ~ ^((Г1/)•(*)) f°r иИ f£S,x€T(k).
(iii) . Let z =[XS]EZ\G, T(k)) be some continuous cocycle. For an arbi-
trary point x G T(k) we denote by m* : к [5] ~>k [5] the homomorphism
corresponding to translation by x:
m : Г® к -+T® к,. m Ay) =xy.
л Л
The cocycle z permits us to define a new action of G on the ring к [5] by
means of the formula
y=w*(s/), sGG, f<Ek[S].
s
Since sm* = m^s, it follows from the ‘cocycle’ condition 5 (x^x^^ == 1 that
this is well defined. The action on к clearly does not change; moreover,
sf= [(sf)(xs)] sf if fes.
(iv) . Now let 5(5) be the subring of invariants in к [5] with respect to the
new action of G. We set Tz = Spec 5(5). The scheme Tz is a principal homo-
geneous space over T, that is, there exists an exterior composition law
TXTZ-*TZ
with the usual properties.
(v) . The principal homogeneous spaces TZz are isomorphic if and
only if Zy, z2 belong to the same cohomology class in H\G, 5). In particular,
the following conditions are equivalent:
(a) z is homologous to zero.
(b) Tz = T.
(c) Tz {k) is non-empty.
This is the usual context in which homogeneous spaces turn up in number
theory: the diophantine property (c) turns out to be equivalent to the coho-
mological condition (a); cf. Cassels [3], where an analogous formalism is de-
scribed for elliptic curves.
30.3.4. A general construction. Let V be some variety over the field k, and let
{Dz} С И® к be a finite set of (irreducible and reduced) divisors. We denote
by 5 the group of principal divisors of which all components belong to ,
and let R be the group of rational functions on V ® к with divisors from 5.
We consider the usual exact sequence
. (IV.18)
It permits us to define a morphism of the complement to U Д into the
torus Spec к [5] in the following way. Because S is free of finite rank, there
exists a section у : S~*R. On the other hand, А С Г(К \ UDZ, where
V- V ® к. Therefore the choice of defines a homomorphism of &-algebras,
fc[5] ->Г(£7, (IV.19)
where U = V \ U£>z*, i.e., a morphism of A:-schemes
£7-> Spec MS] . (IV.20)
We now suppose that the divisor Xz-Dz- is G-invariant. Then (IV.18) is a
sequence of G-modules, and SZ>z- ~D ® k, whereD C Kis some divisor, so
that U - U® к, where U ~ V \ D.
We want to construct a morphism analogous to (IV. 20) over к This can be
done directly if (IV.18) also splits as a sequence of G-modules; then choosing
a G-section ip, we obtain a G-homomorphism of algebras (IV.19) which in-
duces a homomorphism of the rings of invariants
fc[5]G->r(C7, 0и)С = Г(и, 0ц), (IV.21)
and finally a morphism
t/-»Spec£f,S]G = T. (IV.22)
However, in general, the G-section need not exist. The obstructions have
a cohomological character: Consider the exact sequence of G-modules ob-
tained from (IV.18),
1 -> Hom7 (S,k* ) -> Hom7 (S, A) Hom7(5, S) 1 .
Z^ Zj Zj
(Take into account that (IV.18) splits over Z.) The exact cohomology sequence
gives a boundary homomorphism
6 : HomG (5, 5) = #°(G, Homz(5, 5)) ->Я1 (G, Homz(5, I*)) .
The image 6 (id) of the identity map 5 5 is an obstruction to the splitting of
(IV.18). We now remark that according to 30.3.3(ii), Homz(^ k*)= Г(£) (as
G-modules), where T = Spec к [s]G. Therefore the obstruction written down
defines at the same time some principal homogeneous space Tz over T (see
30.3.3 (iv), (v)). It appears that in the general case one only needs to replace
Tby it in the diagram (IV. 22).
Proposition 30.3.5. With the notation of the previous paragraph, there exists
a cocycle z in the class b (id) E (G, T(k)) such that it is possible to con-
struct a k-morphism
U-*TZ (IV.23)
such that over к it is isomorphic to diagram (IV.20).
Proof. We proceed by an explicit calculation. To find a cocycle in the class
5 (id) we need first to construct some inverse image of id in Hom (5, R). This
is also a section ip : S-+R, After this, the formulae
= f^S, s&G, (IV.24)
yield a cocycle
{zjez'fG, Hom (5, fc*)) ,
1J
the class of which also belongs to 6 (id). Since the map S -+ k* : g (s™1#)
(for a fixed s E G) is a homomorphism, there exists also a uniquely defined
point x5. E T(k) such that
2s(/) = (tf)(xs), sGG. (IV.25)
From formula (IV.24) one easily deduces the explicit form of the condition
that{zs} be a cocycle;
s(zf(/))z (?f)z (/) 1 = 1 .
Substituting formula (IV.25) in this we obtain that for all/G S,
(stfy [sxt-xs -х~1] = 1 ,
that is, Т(кУ). The class of {xs} coincides with 6 (id) under the
identification of Hom (5, к * ) with T(k).
On the other hand, comparing formulae (IV.24) and (IV.25) gives for
G S,
^(/)) = (sf) (-S) Ш)]. (IV.26)
This equality can be interpreted as follows. We introduce on к [5] a new action
of the group G : sf, by the formula sf= m*$ (sf), as in 30.3.3 (iii). Then
extends to a G-homomorphism of A;-algebras
k[S] ->Г((7, 0^). (IV.27)
Passing to the spectra of the subrings of invariant elements, we finally obtain
the desired morphism.
U~»TZ ,
which concludes the proof.
1 30.3.6. Proof of Theorem 303J. We apply the construction of 30.3.4 and 30.3.5
to the case where К is a Del Pezzo surface of degree 6 and {Dj} G V ® к the
set of all exceptional curves on it. We shall show that in this case the map (IV.22)
is an isomorphism. For this it suffices to verify that it becomes an isomorphism
after replacing A; by k. But over к all Del Pezzo surfaces of degree 6 are iso-
morphic. Therefore we can carry out all calculations for one suitable model.
For this, we choose the surface Vq C pl X P1 X pl given by the equation
(IV.28)
where (Tq, Tj), (Tq, ), (Tq, ) are homogeneous coordinates for the
first, second and third factor, respectively. To identify Vf it suffices to observe
that the projection p^ : f ^Pl X P1 presents К as a monoidal transformation
with as centre the pair of points ((0,1), (1,0)) and ((l,0),(0,1)).
Using the notation of Section 30.3.4, the group R is generated by k* and the
functions Tg/Tp Гд/Гр restricted to V. The groups is free.of rank
two (in virtue of relation (IV.28)), so that the torus Spec к [5] is two-dimen-
sional. One immediately verifies that the canonical map
U=V^D^ Spec & [T0/Tp T\/TQ, T[/T^]
is an isomorphism. This proves the theorem.
30.3.7. Proof of Theorem 30.3.2. Let Tbe a torus over a number field к and
let K/k be a finite normal extension over which it splits, that is, T becomes
isomorphic to K over K.
The deviation from the Hasse principle in the class of principal homoge-
neous spaces over T is measured by the group
Ker [H^G, T(K)) nu^(G , T(Kv))] ,
where v runs through all possible valuations of k, and Gv C G - Gal (K/k) is
the decomposition group of some extension of v on К In fact, the non-zero
elements of this kernel correspond to those spaces which have a point every-
where locally, but not globally. One can restrict oneself to the field К instead
of к because for a trivial torus the group Hfi is also trivial. The order of this
kernel will be denoted by z(T).
The Hasse principle holds if and only if i(T)= 1. Ono [2] proves that this
equality is satisfied in the following two cases:
(i) Suppose the torus Tsplits over a cyclic extension of the field k\ then
i(T) - 1. This is Proposition 4.5.1 of Ono [2].
(ii) Let H2(G, T) = 0, where t is the group of characters of T\ then
i(T) ~ 1. In fact, let T(AK) be the group of adeles of the torus T, and C ~
~ T(A^)/T(Kf The exact sequence
Z7°(G, C) -+ Hl(G, T(K)) H1 (G, Т(А^))
shows that Ker = 0 if C) = 0. By Nakayama’s duality theorem (see
Ono [2], Section 2.2)),Z/°(G, C) t). Therefore Ker 0 in our
case. But Ker coincides with the kernel of ЯЦС, T (К)) -> T(KV)).
These two criteria can be combined by means of the following functorial
properties of the number i(T):
(iii) i(I\ X T2) = /(Ti) • i(T2);i(RK/k(Ty) = i(R^k(T)) = i(T).
Here Rfc]k denotes the Weil restriction functor (to the base field) and R oXT)
is the kernel of the norm map. Using these criteria, Voskresenskil establishes
that i(T) = 1 for all two-dimensional tori Г by simply sorting out all cases.
We describe this in a few words.
We can assume that the representation G Aut (T) = GL(2, Z) is faithful.
Its image is a finite subgroup in GL(2, Z), which is defined up to conjugacy.
There are in all fifteen classes of such subgroups: all of them are mentioned
in the paper of Voskresenskil [1]. The intersection of any such subgroup with
SL(2, Z) is cyclic: every finite subgroup of SL(2, Z) is isomorphic with (1),
Z2>Z3>Z4 or Z6. Because criterion (i) immediately gives what we want for
cyclic groups, we need only consider those cases where the image of G is not
completely contained in SL(2, Z).
There are eight such classes. Criteria (iii) and (i) give the desired result for
all except two of them. There remain the cases G = JV(A3), the most inter-
esting case for us, because only this one (except G = Z6) can furnish a mini-
mal Del Pezzo surface. Therefore we shall analyze this one in more detail
(these are the groups 9a, b in the list of Voskresenskil).
(a) G is generated by (j ”*), (J *) in GL(2, Z), f - Z X Z, in the natural
representation. Let G2 and G3 be the respective Sylow subgroups of exponent
2 and 3 of the group G. To show that H\G, T ) = 0, it suffices to check this
for G2, G3. But G3 — Z3 and G2 = Z2 X Z2, and a direct, simple calcu-
lation gives the desired result.
(b) G is generated by (| ~l), (^ q1) in GL(2, Z), T = Z X Z. Here the
analysis is similar.
This concludes the proof.
As a corollary we indicate a class of minimal cubic surfaces for which the
Hasse principle is still correct:
Theorem 30.3.8. The Hasse principle holds for those minimal cubic sur-
faces over a number field к which become non-minimal under some quadratic ex-
tension.
Proof. Let V be such a surface, and let H C G = Gal (к/к) be the subgroup
of index 2 corresponding to a field К over which V becomes non-minimal. We
write G = H U sH. Let D G Div (K ® k) be a sum of exceptional curves which
are conjugate over К such that this sum can be collapsed over K. Then the
divisor/) + sD is G-invariant. Because V is minimal, we have according to
Theorem 28.1,
0v(P + sD)^n~y, y>l.
On the other hand, let D consist of x components each of which intersects n
components of sD, We have first of all:
(D + sD, S2"1) = 3y = 2x .
Consequently, x = 0 (mod 3); but x < 6 and therefore x = 3 or x - 6. Further,
(D+sD, D + sD) = Зу2 = (2л - 2)x.
Therefore there^are two possible cases:
(i) x - 3, j - 2, n = 3. The graph of the orbit D + s£> is
Here every vertex is joined with five of the vertices of the other row — with all
except the opposite one. In the picture we have shown only five of these sim-
plexes. This configuration is classically called a ‘conjugate Schlaffli sixfold’.
Case (i) is realized by means of a cyclic subgroup in W(/?6): its generator
belongs to class 5 in Table 1 of Section 31. For case (ii), the cyclic subgroups
are not sufficient; we leave it to the reader as an exercise to check that it can
be realised.
We now suppose that the set K(/cv) is non-empty for all v. Then also K(A?V)
is non-empty for all v. Because the index of К® K, as shown, is equal to 3 or
6, it follows from Theorem 30.3 (ii) that the set K(A?) is non-empty. Let
x G F(K), and let x be the conjugate point of x over k. Then x о x' G V(k).
This proves the theorem.
Problem 30.4. Is it true that every cubic surface over Q has points over
some Abelian extension of Q? This is the simplest partial case of the Artin
hypothesis, to the effect that the maximal Abelian extension of Q is a
Q-field (see Greenberg [1]).
31. Tables and comments. Calculation of Як The theorem of Artin and Tate
In this section (pp. 176— 178) there are three tables. Below is explained
their structure, possible use and method of calculation. Cf. also the com-
ments in Section 23.11. After describing the tables we mention one general
lemma on the calculation of Яг(6, Я (К)) and the theorem of Artin and
Tate which connects the order of Я1 with other invariants of the represen-
tation.
31.1. Table 1. This is a table of the classes of conjugate elements in the
Weyl group И\Я6) = JV(£6). It is obtained by slightly revising the tables of
Swinnerton-Dyer [2]. Needless to say that all responsibility for mistakes is
the author’s.
(1) . The first column of the table gives the number of the class. It does not
coincide with the Swinnerton-Dyer number, the designation of which (of
type Q) is shown in the zeroth column.
(2) . The second column gives the value of the index. For the definition,
see Section 28.2. We recall that the index is calculated in terms of the action
of KV^) on the set of exceptional classes Z6. If the corresponding element
of the group is realized as an automorphism of a cyclic extension K/k and acts
on 7V(F) - Pic (К ® K) for a cubic surface K, then the index is the maximal
number of lines on V which can be simultaneously collapsed over k. In par-
ticular, the first five classes correspond to ^-minimal models, and the following
six to surfaces on which only one line can be collapsed, etc.
(3) . The third column gives the order of an element in the class.
(4) . The fourth column (measure™1) gives the quotient of the order of the
group and the number of elements of the class. The name is connected with
the Cebotarev -Artin density law; see Serre [7].
Suppose that К is a cubic surface over a number field к and that the image
of G = Gal (k/k) coincides with the whole group PF(F6). Then the measure
of the class of conjugate elements is equal to the Dirichlet density of those
prime ideals of the field к for which the corresponding Frobenius automor-
phism falls into that class (see Section 27).
(5) . The symbol ambncP ... in the fifth column means that in the canoni-
cal representation of JV(R 6) on an element from the class under considera-
tion has as eigenvalues m primitive roots of unity,?? primitive 6th roots of
unity, etc. For surfaces over a finite field of q elements, this column permits
us to calculate the zeta function if the class of the Frobenius automorphism
is known.
(6) . The trace, which is mentioned in the sixth column, gives directly the
number of points; see Theorem 27.1 and its collaries.
(7) . The seventh column contains the values of/f^G, where G is the
cyclic subgroup in W(R6) generated by some element of the class. Its algebraic-
geometric meaning has in part already been described. Some information on
the method of calculating it is given below.
Finally, the eighth and ninth columns yield information on the action of the
cyclic subgroup G on 76.
(8) . In the eighth column, the symbol ambn ... means that there are m or-
bits of order a, n orbits of order b, etc.. A notation of the type 63 6 means
that of four orbits of six lines, three have one type of connectivity and one
another. The type of connectivity is given by the matrix of the pairwise inter-
section numbers and it is not explicitly given. It can be derived from the
second table of Swinnerton-Dyer [2], where all orbits are listed.
(9) The roman numerals in the ninth column for the classes 6—11 refer to
Table 2. There the numbers of the decompositions into orbits (with respect to
G) of the graph Г5 are shown; this graph consists of the vertices in Г6 which
are not joined to some G-invariant vertex.
For classes 15—25 we show the type of the permutation which is induced
on the ^-exceptional subset of length 6 in /$. Similar information is given for
the classes 12 —14; see the explanations in Sections 28.2—28.4.
Table 1
Classes of conjugate elements in the Weyl group ГР(7?6) = W(£6). For an explanation, see text.
0 Swinnerton-Dyer numbering 1 No. of the class 2 Index 3 Order 4 Measure”1 5 Characteristic roots on Pic 6 Trace on Pic 7 Я1 8 Number of orbits 9 Type of decomposition
C13 1 0 12 12 132 124 0 0 3-122
^12 2 0 6 72 132 64 2 0 3-64
3 0 3 648 136 -2 Z3XZ3 39
^14 4 0 9 9 1-96 1 0 93
^10 5 0 6 36 , 122 32 62 ~1 Z2 X z2 363 6
^24 6 1 12 12 I2 242 62 2 0 1446*12 I
^20 7 1 8 8 1284 1 Z2 1282 8 XVIII
c7 8 1 6 36 13,22 62 2 0 I3 23 63 II
Cj9 9 1 4 96 I2 23 42 z2xz2 123 44 4 X
C4 10 1 4 96 I3 44 3 zo 1346 V
^3 11 1 2 1152 1324 -1 z2xz2 I3 212 IV
^25 12 2 10 10 I2 254 0 0 252 510 ГХ21
0>2 13 3 6 36 I2 234 -1 0 33 32 62 SX31
cs 14 3 6 24 I3 22 32 0 0 122 22 32 62 sxl’21
^23 15 6 6 12 l2 232 62 1 0 362 66 61
C1S 16 6 5 10 1354 : 2 0 I2 52 52 5 I1 51
THE TWENTY-SEVEN LINES CH. IV, § 31
Table 1 (continued)
0 1 2 3 . 4 5 6 7 8 9
Swinnerton-Dyer numbering No. of the class Index Order Measure”1 Characteristic roots on Pic Trace on Pic Я1 Number of orbits Type of decomposition
c5 17 6 4 16 l3 22 42 1 0 122 242 42 4 24
c9 18 6 3 108 I3 34 1 0 36 З3 32
^18 19 6 4 32 I4 242 3 0 Is 244 4 I2 4
Qzi 20 6 6 36 I4 232 2 0 I3 23 34 6 123
ci7 21 6 2 96 I4 23 1 0 I3 26 26 23
Q 22 6 3 216 I5 32 4 0 1936 I3 3
^2 23 6 2 192 Is 22 3 0 I7 28 22 I2 22
C16 24 6 2 1440 I6 2 5 0 11S 26 I4 2
Cl 25 6 1 51840 I7 7 0 I27 I6
сн: IV, § 31 TABLES AND COMMENTS, CALCULATION OF H
W 2Х2Г ~XVTT 2XVUI XDC
Table 3
№ Type of decomposition
0 I, II, III, V, XII, XIII, XIV, XV, XVIII, XIX
Z2 VI, VII, VIII, IX, XI, XVI, XVII
Zj X Z2 IV, X
31.2. Tables 2 and 3. Let H C W(#5) be a subgroup such that under its action
on/5 no orbit is an exceptional set. When interpreted algebraic—geometrically,
these subgroups correspond to minimal Del Pezzo surfaces of degree 4.
H is some subgroup of the group of automorphisms of the graph Г5 de-
scribed in Section 26.9, and it therefore decomposes the set of its vertices into
orbits. Table 2 contains all possible decompositions of this kind (to the cyclic
subgroups correspond only those five of them that are shown in column
nine of Table 1). In Table 3 the values of the groups are given for
each of these decompositions. The circumstance that#1 depends only on the
decomposition and not on the group H itself is not entirely obvious. This will
follow from Proposition 31.3 below, which also gives a convenient direct
method of calculating /71.
Tables 2 and 3 are taken from Manin [4],. which contains a detailed proof
of the completeness of the list of decompositions. Originally, these data were
calculated in order to obtain an, albeit partial, birational classification of cubic
surfaces with one rational line.
The fact is that cubic surfaces of index > 2 with a non-empty set of points
are birationally trivial (Theorem 30.3), and for surfaces of index 0, the bira-
tional classification coincides with the projective one (see Chapter V). Neither
the one nor the other is true for surfaces of index 1. For this transitional case
there are no decisive results, and Я1 is the only accessible birational invariant.
Proposition 31.3. Let V be a smooth projective surface over a field k, and
let {ty} С V ® к be a finite G-invariant set of irreducible curves the classes
of which generate N(V\ Let S be the group of divisors generated by {#/},
let Sq CS be the subgroup of principal divisors, and let H C G be a normal
subgroup of finite index which acts trivially on allD^ Finally we set
W=( Zz jleZ[G/ff|.
Then
Hr(G, N(V) =* (NS П S0)/NS0 .
(The isomorphism is not canonical.}
Proof. First of all, H'(G, N(V)) = HX(G'H, 7V(E)) (inflation). The exact
sequence of G/H modules
180 THE TWENTY-SEVEN LINES CH; IV, §31
6-*^ ->S -> Pic(K® K)-> 0, K = kH ,
gives
0 -*H-X(G/H, Pic(V® K))-*H°(G/H, SQ)-+H°(G/H, S),
from which
H~x (G/H, Pic (V ® К )) “ Ker (5^/NS() -> SG/NS) = (NS П SQ)/NSQ .
It now remains for us to observe that the canonical pairing
Pic(7®/QX Pic(7® /Q~>Z
(the intersection number) induces a pairing of cohomology groups
HP(G/H, Pic (V ® К)) X H~P(G/H, Pic (V ® K)) ->lfi(G/H, Z) ,
which is an exact duality. The proof of this is a slight generalization of the
argument of Theorem 6.6, Chapter XII, in the book of Cartan—Eilenberg.
As Pic (7 ® K) is a free group of finite rank, the intersection number per-
mits us to identify Pic (V ® K) and Hom (Pic (И ® K), Z), which gives an
exact sequence of G/H modules,
0 Pic (И ® K) -+ Hom (Pic (7 ® X), Q) Hom (Pic(7 ® K), Q/Z) 0 .
Its boundary isomorphism
5 : flP(G/ff,Hom(Pic(F® K),Q/Z))=;flP+1(G//f, Pic(7® K))
can be combined with the Cartan—Eilenberg isomorphism, from which
Hom Pic (И® X)), Q/Z) HP(G/H, Pic (7 ® X)) .
We omit the easy verification that this pairing is the same one as the one de-
fined earlier. This proves the proposition.
3\ A. Application to the calculation of Hf Let V be a Del Pezzo surface
of degree d < 7. For{D;Jwe take the set of all exceptional curves on V ® k.
We identify this set with the vertices of the graph Гг,г = 9 — d. Then S is
transformed into the group of zero-dimensional chains of this graph, and
into the subgroup of chains SafZz- for which Saz (Zz-, Zy) = 0 for all Zy G (vertices
of ГД Further, let (ZK) C 5 be the set of those chains which are sums of the
vertices of some Я-orbit, where H is the image of the Galois group in Aut Гг
We denote by the number of vertices in Zv, and by h the order of Я. Then:
(i) The group TVS' is generated by the chains (A/zK) ZK = norm of a vertex
ofZK.
(ii) The group TVSq consists of the chains
{Sa/A/z^ZjV/, SaK(ft/zK)(ZK,Z.) = 0}.
Here everything, except the number A, depends only on the decomposition of
Гг into orbits. But this factor also ceases to play a role after passing to the
quotient group (TVS’ П S'q)/TVS’o : by the linearity for all h = 0 (mod zK), we
obtain one and the same value, and in practice it is convenient to choose
l.c.m. (zK) instead of h in (i) and (ii).
In this way Table 3 has been calculated for r ~ 5.
We cite, finally, a theorem of M. Artin and J. Tate.
Theorem 31.5. Let V be a smooth projective surface over a finite field k,
and let V ® к be birationally trivial. We denote by the order of the group
H\G, TV(F)), where G - Gal (k/k),and by (f) the set of characteristic roots
different from unity of the (algebraic) Frobenius automorphism acting on
N(V). Finally, let A be the determinant of the form 'intersection number’ on
N(V)g. Then
П(1-г) = л1д.
We omit the proof. This is a particular case of a considerably more general
fact which is established in Tate’s paper [1] not only for birationally trivial
surfaces. (In the paper this equality is only proved up to a factor which is a
power of the characteristic of the base field; but here this restriction can be
removed.) See also Milne [1].
The reader himself can verify as an easy exercise the next proposition.
Proposition 31.6. Let V’, V satisfy the conditions of Theorem 31.5, and
let there exist some birational morphism V' V. Then Theorem 31.5 holds
for V' if and only if it holds for V,
This reduces the problem to minimal surfaces К Applying this result to
Del Pezzo surfaces, and translating it in terms of our model with Weyl groups,
we obtain:
Conjecture 31.7. Let 3 < r < 8 and let H C JV(Ar) be some cyclic subgroup
of the Weyl group of the root system Rr. We denote by /г1 the order of the group
Nr), by (f) the set of characteristic roots different from unity of a
generator of the group H acting on N and by A the determinant of the met-
ric on Then
П(1-?)=л1д.
r
I do not know a direct algebraic proof of this result; of course, it is easily
checked for by means of the tables.
Problem 31.8. Does 31.7 generalize to the remaining Weyl groups?
32. Bibliographical remarks
Zeta functions of rational surfaces were computed by Weil [2]. He also
gave an outline of the applications of cohomology theory and the Lefshetz
formulae (the Weil conjectures). The detailed formalism is developed in the
recent paper by Kleiman [1].
The group of automorphisms of the set of twenty-seven lines has been
studied by Jordan [1]; see also the bibliographies in Henderson [1], Segre [1]
and Swinnerton-Dyer [2]. The treatment given here is based on Manin [3],
with improvements by P. Deligne. It is interesting to note that the simple roots
in the system Rr get an algebraic-geometrical meaning if Nr is realized as
the group 7V(K) for some ‘degenerate’ Del Pezzo surface V. A map on it by
means of STy1 collapses some configurations of curves with self-intersection
number —2 into ‘rational’ singular points. The classes of these curves precisely
constitute a system of simple roots; see Lipman [1 ]. The proof of Theorem
24.5 has been taken from Safarevi£ [1]. The cohomological invariant was
introduced in Manin [4]; its connection with degrees of unirationality is ex-
plained in Manin [3].
The twenty-seven lines were discovered in 1849 by Salmon and Cayley. A
detailed bibliography of the study of them in the first 50 years is contained,
in Henderson [1] and in the survey of Meyer [1]. We cite some fragments of
Henderson’s historical summary:
“Indeed Sylvester once remarked, in his characteristical florid style:
‘Surely with as good reason as had Archimedes to have the cylinder, cone and
sphere engraved on his tombstone might our distinguished countrymen leave
testamentary directions for the cubic eikosiheptagram to be engraved on theirs.”
“If Cayley and Salmon had wished to follow Sylvester’s advice and to insert
a clause in their wills, directing.that an eikosiheptagram be engraved upon
their monuments, they would have had no. certainty of the correct fulfilment
of their directions until the year 1869, when Christian Wiener made a model
of a cubic surface showing twenty-seven real lines lying upon it. This achieve-
ment of Wiener, Sylvester once remarked, is one of the discoveries ‘which
must for ever make 1869 stand out in the Annals of Science’.”
“Klein exhibited a complete set of models of cubic surface^ at the World’s
Exposition in Chicago in 1894, including Clebsch’s symmetrical model of the
diagonal surface and Klein’s model of the cubic surface having four real coni-
cal points. Models of the typical cases of all the principal forms of cubic sur-
faces have been constructed by Rodenberg for Brill’s collection; and these
plaster models may now be purchased.”
Henderson’s book appeared in 1911. Sylvester’s eloquence and the plaster
models disappeared together with the Victorian era ... .
CHAPTER V
MINIMAL CUBIC SURFACES
33. A survey of the results
In this chapter, the base field к is perfect. We shall investigate the category
of minimal smooth cubic surfaces over к and birational maps, and prove the
following basic facts about them.
Theorem 33 Л. Every minimal cubic surface over к is birationally non-triv-
ial,.
Theorem 33.2. Two minimal cubic surfaces F), over к are birationally
equivalent if and only if they are (protectively) isomorphic.
Hence it follows that all linear constructions with minimal surfaces have a
birationally invariant meaning. In particular the following corollary holds.
Corollary 33.3. If Kp are birationally equivalent, then the commutative
Moufang loops Vy(k)!Uf V^fktjU are isomorphic (U is the universal equiva-
lence relation', see Section 11).
We note that Theorem 33.2 is certainly not true for non-minimal cubic
surfaces. In particular, it does not say anything for algebraically closed fields.
Example 33.4. Let Va be given as in Example 23.4 by the equation
J’S + tI + tI + aTl = 0 .
If a = bc$, then Va and are isomorphic: simply substitute = cT^ in the
equation of Va.
Exercise 33.5. If a £b(k*)\ then Va and Vb are not isomorphic and there-
fore not birationally equivalent.
Hint. (i). As in Example 23.4, it suffices to show that Va and И4 are not
isomorphic.
(ii). Every projective isomorphism of Va and Vb must induce an isomor-
phism of the sets of the 27 lines.
33.6. To describe the birational automorphisms of V, we need an additional
construction.
In this chapter we shall call a point x G V(k) good if it does not lie on the
union of the lines of V ® k. An unordered pair of points x, у G V(k) is called
good if x Ф у and the line containing these points is not tangent to V ® к and
does not intersect the union of the lines on V ® к in P|. The point x оу =
= is then also good, as are all pairs from (x, y, x oy).
The following identity holds:
t t t ~ t t t .
x хоу у у xoy X
(V.l)
In fact,*, x о у, у and a general point v G Г lie in one plane; the intersection
with this plane is a cubic curve, and on this curve (tx txoy ty)^v = v (see Theo-
rem 2.1 (iii)).
Now suppose that the points x, у form a good pair and that they are defined
and conjugate over a quadratic extension К Э к. The birational automorphism
txtXQy ty *s ^еп also defined over A", and, as an automorphism of the function
field on V ® K, it commutes with the conjugation of K/k, acting trivially on V.
Galois theory then immediately implies that there exists a birational
automorphism sxy of the surface V over к such that
sx,y® K 1хгхоу{у-
(V.2)
We are now in a position to state theorems on the structure of the birational
automorphisms.
Theorem 33.7. Let W be the group of projective k-automorphisms of a
minimal cubic surface K, and let В be the group of birational maps generated
by the maps tx for all good points x G V(k) and the maps sx,y for all good
pairs x, у G V(k) which are defined and conjugate over a quadratic extension
of the field k. Then the full group Bir V of birational maps V -> V over к is
generated by the subgroups W and B.
Thus we have described the generators of the group Bit V, We note that W
is finite'. its representation on the graph of the lines of V ® к is faithful be-
cause the kernel of the representation acts trivially on the lines and hence its
elements arise from the automorphisms of the projective plane which leave six
points fixed. Except for the identity, there are no such automorphisms.
On the contrary, the group В is very big, provided that F(fc) is non-empty.
Meanwhile it is clear that it has many generators: the set {fj indexed by the
points of И(к) (outside the lines), and the set{sx indexed by the pairs (a
point x cy of K(k); a line in P3 which is rational over к and which intersects
V in good points that are not defined over k). Thus the set of generators is
something like U(£) X P2(fc).
It turns out that there are not that many relations between them.
Theorem 33.8. All relations between the generators of the group BW are
consequences of the following ones'.
t2x-^z = \, wtxw~x=tw(x)>
(V.3)
(tX‘xou = 1 ’ WSy,ZW“1 = Sw(y),w(z)
In particular, BW is a semidirect product (with В as the normal subgroup).
We deduce two corollaries.
Corollary 33.8.1. Let Vbea minimal cubic surface over a field к with a suf-
ficiently large number of elements (infinitely many, for example). Then the
following assertions are equivalent'.
(i) The set V(k) is non-empty.
(ii) The group Bir V of birational к-maps V -> V contains not only pro-
jective maps.
(iii) The group Bir V is infinite.
Proof, (i) => (ii). It follows from Theorem 30.1 that there is a good point
x G И(&); the map tx is then not projective.
(ii) => (i). It is clear from Theorem 33.7 that if the birational maps of V are
not exhausted by the projective ones, then there exist maps of the form tx or
Sy z, with respectively x G 7(k) or у о z E V(k).
(i) => (iii). According to Theorems 29.4 and 30.1, the set L(&) is big (it is
dense if к is infinite). Therefore there exists a go jd pair of points x, у G L(Z).
One can then conclude from the relations (V 31 that the map tx ty in the
group BW has infinite order. It is even simpler to prove this by means of some
intermediate results.(see Example 39.8.4 below).
(iii) => (ii). This has already been proved.
33.8.2 . The second corollary permits us to say something about the struc-
ture of the group В when comparing it to the orthogonal groups of rank three.
We shall assume that the characteristic of к is not equal to two. Let L be
some и-dimensional linear space over £, let /: Z к be a non-degenerate qua-
dratic form on £, and let b : L X L к be the corresponding bilinear form
Let On{f) be the group of orthogonal automorphisms with respect to/
over k. A reflection tx
r (v)=y_bl^ X,
y f(x) ’
is associated with every vector x EL for which/(x) Ф 0 (see E. Artin [1]).
These reflections satisfy the relations
^=1’ (TXTJTZ)2 = 1 (V-4)
if x, y, z are linearly dependent. It is trivial to verify this. [As a matter of fact,
the rx generate ОД/), and a complete system of relations looks like tx = 1,
rx Ty Tz ~Tu’ w^ere u can be explicitly expressed in terms of x, y, z. This is
proved in Becken [1], but we shall not need it.]
Corollary 33.8.3. Let V С P3 be a minimal cubic surface over the field k.
Let us choose'.
(i) an arbitrary plane P2 С P3 over k\
(ii) an arbitrary point x0 G P3(k) which is not in V U P2;
(iii) an arbitrary non-degenerate quadratic form f on the linear space
Z = P3\P2 which is zero at xQ,.and for which there are no isotropic
points on V(i.e., /(x) Ф 0 for allx G F(k) except the points at infinity
ои(ГПР2)(к)).
Then the map
extends to a homomorphism of the subgroup Bq C B, generated by the tx
for all good points x G V(kf into the group 0^(f).
Proof. One only needs to compare the relations (V.3) and (V.4) and to ob-
serve that the vectors in L corresponding to the points x, x by andj on V(k)
are linearly dependent.
We note that this proof has a purely algebraic character and essentially uses
the fact that the system of relations (V.3) is complete. The geometric meaning'
of the homomorphism described above is quite vague: the groups Bq and O^(f)
have an entirely different nature, and the possibility of largely varying f looks
strange.
Another astonishing property of the group of birational automorphisms of
V is that В is a normal subgroup in it. I do not know any natural representa-
tion of this group with kernel B.
One would like to use Theorem 33.8 to obtain further information on the
group BW. For instance, I do not know the answer to the following problem.
Problem 33.8.4. Does В contain elements of finite order other than tx, sy,z
and their conjugates?
Maybe the homomorphisms of Corollary 33.8.3 are useful here.
For a better understanding of the structure of B, one may consider the sub-
groups generated by those tx for which x belongs to some hyperplane section.
A pilot problem: H is some Abelian group (cyclic in the simplest case). Examine
the group B(H) generated by the generators ta (for all а ЕЯ) with the rela-
tionsr2 = (rar&r_e_d)2 = l.
33.9. Remarks on the method of the proof In Chapter IV we used the
representation of Gal (k/k) on the object {Af(F), intersection number}.
The group of birational maps of V does not, generally speaking, act biregu-
larly on any model of the function field on V: to resolve the singularities of
each given map, we must blow up some points of V, depending on this map.
A reckless solution is to blow up all points of all models^ it turns out that this
can be carried out reasonably well, and one obtains an algebraic object which
here will be called ‘the fundamental birational invariant’ and will be described
in the next section. It looks like {lim7V(K')} Inn lim (intersection num-
ber), ...} over all birational morphisms F* -> V. This monster has been tamed
by Max Noether and the classical Italian mathematicians; the contemporary
authors have only braided ribbons into its tail (cf. Deligne [11).
33.10. Comparison with other dimensions. If one replaces the words ‘mini-
mal cubic surface’ in Theorems 33.1, 33.2 and 33.7 by the words ‘smooth
plane cubic curve having a к-point’, then the assertions remain true, although
the mechanism of the proof in the one-dimensional case is quite different.
However, the group of automorphisms of a cubic curve V with an empty set
of points R(k) cannot be exhausted by the projective automorphisms, because
K0(k) acts on K, where F$ is the Jacobian of V (see Cassels [1]). Apparently
this fact has no analogues in the two-dimensional case, but even the existing
parallelism is astonishing. On the other hand, for three-dimensional cubic hyper-
surfaces (let us say over C) the truth of the analogues of Theorems 33.1 and
33.2 would mean a solution to an old problem.
These problems have been neglected because they turned out to be very diffi-
cult. Fano has spended some decades on experimenting and on carrying over
the methods of two-dimensional birational geometry to the three-dimension-
al case. His series of papers [1 ] —[4] and others (see also the bibliography
in Roth [1]) contains a mass of interesting calculations but no coherent proofs
which can stand up to criticism.
Maybe the technique developed in this chapter can be carried over to higher
dimensions. The formal constructions, with the replacement of 7V(F) by the
Chow ring, may lead us far enough; all the material of Section 34 has
a reasonable generalization. Here the Todd genus and the calculations of the
Chow ring of a monoidal transformation by Grothendieck are used. The diffi-
culties start in Section 35, especially towards the end.
34. The fundamental birational invariant
In Chapter IV an important role was played by the triple {7V(K), со у, in-
tersection number on Af(K)}, which can be constructed for every surface V.
In particular, the action of the Galois group and the group of automorphisms
of V on this triple yields essential invariants of the surface.
In this section we shall construct an ‘infinite’ analogue of this object which
is suitable for studying the birational maps of V.
We first collect the necessary facts from Chapter Ш. Recall that Pic V is at
the same time a covariant and a contravariant functor on the category of sur-
faces with birational morphisms/: V' -> V. The corresponding maps of the
Picard groups are denoted by / and/*. The composition law in Pic V is
written additively. We denote the class of the canonical sheaf by со у GPic V.
Lemma 34.1. Let f'. V1 V be a birational morphism,
(i) /* : Pic VPic Vf is a monomorphism which preserves the intersection
number,
(И) АГ = id;Pic Vf = Im/* © Ker/, .
(iii) (/,(/)„/) = (/',/*(/))/oraZZZ'GPic V', ZG Pic V, In particular, Im/*
and Ker /, are orthogonal with respect to the intersection number.
(iv) /*(wK') =
This result is easily obtained by induction on the index of / (the number of
monoidal transformations into which / decomposes) from the following more
exact lemma which only deals with monoidal transformations.
Lemma 34.2. Let /: Vr -> V be a monoidal transformation with a closed
point x E Vas centre. We set d = (k(x): k) and D ~f~^ (x). Let IE Pic V’ be
the class of the sheaf Oy (Df Then :
(i)Ker/, = ZZ.
(ii)(Z, Z) = — d.
(iii) wr,=r*(wK) + /.
In the following we shall most of all work with the group Pic (V ® k)9
which will be denoted by 7V(U). Lemmas 34.1 and 34.2 can be applied to this
group, making the obvious changes. It carries the following family of struc-
tures.
34.3. (i). The Galois group G - Gal (k/k) acts linearly on 7V(K) through the
second factor of V ® k.
(ii) . The intersection number defines a G-invariant bilinear form
7V(K) X 7V(K) -> Z.
(iii) . There is a distinguished G-invariant semigroup 7V+(K) = {/ G/V/U) 11 is th
class of a sheaf L such that dim ® k, L) > 0} (In other words,con-
sists of the classes of effective divisors.)
(iv) . The dual cone2V+(K) ={Z G7V(K)I (/, /')> 0 for all l'eN+(V)}. (It
does not necessarily coincide with7V+: the exceptional classes, for instance,
belong to but not to 7V+.)
(v) . A distinguished element cvy G7V(K).
Now let f: F' -> Ube some A>morphism. We shall for brevity write/* in-
stead of (f®k )* and so on. Then:
(vi) . /*(7V+(U)) C 7V+(U'), 4(7V+(K')) G 7V+(K). (This is obvious from the
explicit construction of the action of f* and/* on divisors: their effectivity
is preserved.)
(vii) . C 7V+(U'). (This follows from Lemma 34.1 (iii) and the
preceding remarks).
34.4. We shall now prepare the necessary data for passing to the limit. Let
К D к be some field and let B(K® K) be the category of which the objects
are birational A^-morphisms V' V ® К (where is a complete smooth
^-surface). The morphisms in this category are the commutative triangles
For any two objects of B(U® K) the set of morphisms is either empty or it
consists of precisely one element. Therefore we shall often identify isomorphic
objects.
The set of objects of B(U® K) is naturally ordered by the relation: />g
(fdominates g) if Hom (/ g) is non-empty. The theorem on resolution of singu-
larities (Theorem 18.4) shows that this set is directed: for every two objects.
gp g2 there exists a third object/such that/>gb In fact, let
gj : Vx V®K, g2 : V2 -> V®K, and let hi: V3 -> Uz- (z = 1,2) be the resolu-
tion of singularities of the map g21 ° gp Then /= gj Q // - g2 ° ^72 dominates
gj and g2.
Lemma 34.5. Replacing the base field к by К transforms B(U) into a
cofinal subset ofB(V®K)if the extension К D к is algebraic.
An informal explanation: Suppose we have, say, a projective system of
groups, indexed by the ordered set of birational mo- Jrisms V’ V
Then we can take the limit, while restricting ourselves to those morphisms
which are defined over k.
Proof of Lemma 34.5. We must check that for every /Gmorphism
/: V’ -> V ® К there exists a й-morphism g: Vn -+ V such that g ® A domi-
nates f. The idea of the proof is the following: Include the points of indeter-
minacy of (/® k)"1 on V ® к in a G-invariant finite set, after having added
all conjugates, and blow up this larger set; this can be done over k.
To carry out this idea, however, it is necessary to take care separately of
the case when there are among the points of indeterminacy of /-1 ‘infinitely
close points’ in the Italian terminology. In other words, in the decomposition
f~fn • • • fi hito monoidal transformations, the centre of fi can lie on a curve
which collapses when, let us say,/-! is applied.
Therefore we first define a canonical decomposition of the morphism
/: V' -> V ® K. Let хг С V ® К be the smallest subscheme of V ® К out-
side which/is an isomorphism. It is closed and zero-dimensional. Let / : -*
К® К be the monoidal transformation with centre Xj (the identity map
if Xj is empty). The proof of the theorem on resolutions shows that/domi-
nates /j. If/-/i, then/ is called the canonical decomposition of/ In the
opposite case, the same construction, applied to I/, gives the subscheme
x2 c ^1 an^ blowing up/2 : whereby/dominates //. Contin-
uing in this way, we obtain a canonical decomposition of/:
f f
V'=Vr
The number r is called the length of the decomposition', we set r = 0 if /is an
isomorphism. It is clear that the length does not exceed the index of/, but it
certainly can be strictly smaller: the length is one if there are no ‘infinitely
close’ points of indeterminacy.
We shall now establish the existence of a morphism g (first paragraph of the
proof) by induction on the length r. If/is an isomorphism, i.e. the decompo-
sition length is zero, then for g we can take V V,
Suppose that everything has been proved for morphisms of decomposition
length r—1. We set /=/'/, where the length of/' is equal to r—1,/':
V ® K. Without loss of generality we can assume that A? is a Galois extension
of the field к with group G.
Letg': U И be a A;-morphism such that#' ® К dominates f. Consider
the commutative diagram
Here/. is the blowing up of a closed reducible subscheme x C This sub-
scheme x can be written as a union x U x", where x". consists of all points in
which h is a local isomorphism. Let и € U ® К be the reducible subscheme
which is the union of all points conjugate over к with points of 7г"1 (x") with
respect to the action of G, There exists a subscheme и G U such that и ® K=u
Let g": V” -> U be the monoidal transformation with centre u. We put
g ~ gg" • V" -> К The morphism g ® К dominates /because h о (g" ® K) is
not a local morphism at any of the points of x and thus it dominates the
monoidal transformation/, with centre x. This proves what we want.
Definition 34.6. Z' (K) = lim N(Vf) and Z.(K) = lim N(Vr). The limits are
taken with respect to the maps /* in the first case, and with respect to the /*
in the second case; V' runs over all possible birational k-morphisms
f: Vf V or к-morphisms V’ -> V ® к: this is the same according to
Lemma 34.5.
Z’(K) plays the role of ‘Picard group of the field of rational functions on
K’. The group Z,(K) will not play any role in the following: it has been intro-
duced for symmetry and some intermediate explanations.
All the structures described in Section 34.3 can be carried over to this in-
finite level (and they shall be used in the following).
34.7. (i). G acts onZ*(K). (Here it is important that we can take the limit
using the k-morphism/: V* -> V, and then /* : Pic (V’ ® k~) Pic (V ® k)
is a G-homomorphism. This also gives the action of G on the limit.)
(ii) . The intersection number furnishes a G-invariant bilinear from
Z* (И) X Ze (K) ^Z (because/* preserves intersection numbers according to
Lemma 34.1 (i)).
(iii) . Z^(K) = lim 7V+(L'). (Use property 34.3 (vi).)
(iv) .Z+*(K) is not introduced by passing to the limit, but by duality:
Z+’(K) = {2 GZ’(K)I (z, z')> 0 for allz'GZ^(K)} .
(v) . £2 : Z’(K) -> Z is defined as the homomorphism wnich on the image
ofjV(K) inZ‘(K) coincides with the intersection number with coyt
^2 (z) — (I, co y)
if I e 7V(K) represents z. The result does not depend on the choice of I by
Lemma 34.1 (iii), (iv).
Thus, there is no ‘canonical class’ in Z’ (K): it belongs to the dual group.
The reason for this is shown by Lemma 34.2: the limit of the canonical classes
would have to coincide with the sum of co у and all exceptional classes in
lim 7V(K)-
More formally, Lemma 34.1 (iv) shows that there exists a limit
П = lim coyt GZt(K), and, using 34.1 (iii), by means of the intersection num-
ber we can define a pairing
Z’(K)XZt(F)->Z
such that S2(z ) - (z, £2) for all z GZ ‘(7). The details are left to the reader as
an exercise.
Theorem 34.8 (Theorem on the fundamental invariant). The composite ob-
ject Z(V) consisting of the groups Z* (K), Z. (K) and the structures described
in 34.7 (i) — (v) is a k-birational invariant of the surface V.
More exactly, the map K*->Z(K) can be extended to a covariant functor
on the category of smooth projective k-surfaces and their birational maps. For
each birational map /: Vf V, the corresponding morphism Z(Vf)~> Z(V)
is an isomorphism.
In particular, the group of birational maps of V into itself is represented in
Z ’ (K) and preserves all the structures described.
Proof. Let/: Vf -+ Vbe a birational map. To define the isomorphism
Z(K') ~>Z(F) (which in the future we shall also denote by /for the sake of
brevity), we note that there are cofinal subsets in В (К') and В (К) which
are identified by means of/. Namely, letg : V" -> V', h : V" Kbe any reso-
lution of the map /. Then B(K") is embedded in В (К') by means of g and
in B(K) by means of h. The images of these inclusions are cofinal.
The independence of this construction from the choice of the resolution
and the functoriality in/are verified immediately and completely standard.
This proves the theorem.
In the following section we shall introduce a certain space which permits
us to indicate a convenient ‘geometric’ system of generators for the group
Z' (K) and to describe the structures introduced above in terms of it.
35. A bubble space
Definition 35.1. E(K) = (U V’)lR, where V' -4 V runs through all objects
of the category В (F), and R is the following equivalence relation:
The points x G V' and x" G V" are equivalent if the canonical birational
map Vf -> V” is an isomorphism of some neighbourhood of xr with some
neighbourhood ofx”.
35.2. Comments. In order to imagine better what the space E(F) looks
like, we shall consider how to glue together with respect to R two surfaces
f: V1 -* V connected by a monoidal transformation with as centre a point
x G V. R identifies V \ x with V' \ D, where D =/"1(x). The closure of
this open set contains the disjoint union/) U{x}. This is a typical case of a
non-separated scheme. If к ~ C, thenD(C) = РЦС) is the Riemann sphere;
thus ((FU Vf)jR) (C) as a topological space is obtained from the four-dimen-
sional space R(C) by attaching a two-dimensional ‘bubble’ Z)(C), blown up
from the point x (see Fig. V.l). Passing to E (K) means that we blow up
bubbles from all points of V, and then from all points of these bubbles, etc.
Fig. V. 1.
35.3. All canonical maps iy>: Vf are open embeddings. We call the
spaces iy(V') the leaves of E(F). Every leaf is dense in E(F). For brevity’s
sake we shall say that a point x G E(V) lies on the surface V1, where
(V’ -> 7) g 8(F), if x lies on the corresponding leaf.
There is a natural ordering on the set of points of E (F): We write x < x"
and we say x lies over x” if there exist surfaces V\ V(> such that xr lies on V',
x” lies on F”, and the canonical map V! -> V” is defined at the point x' and
sends it to x". In other words,x' <x" if x lies on the bubble blown up out
of the point x \ and < is the weakest ordering with this property. (To show
that these definitions are equivalent, one must refer to the proof of the resolu-
tion theorem, as in Lemma 345.) The point x lies over a surface V" if there
exists a point x" on the surface V" for whichx <x".
The geometric points of the space E(F) and the ordering relation on them
are defined analogously.
Definition 35.4. The symbol Z° (F) denotes the free Abelian group gener-
ated by the geometric points of E(F) with values in the field к.
In other words, the elements of Z°(F) are the zero-dimensional cycles the
components of which are not only points of the surface V but also points
lying ‘over’ this surface.
35.5. We equip Z°(F) with the structure of a G-module by using the
natural action of the Galois group G on E(F) (k). We further define a scalar
product
Z°(7)XZ°(7)^Z
by setting (x, = — 1 and (x, у) = 0 ifx Фу, for all x, у G E(F)(k).
We now consider the orthogonal direct product 7V(F) X Z°(F) and define
a homomorphism
7V(F)XZ°(K)-*Z'(y)
as follows. On the first factor it is the canonical map of an element of the in-
ductive system of groups W(F')} in its limit.
Now letx GE(F)(k), and let x lie on the surface F', (Vf -> F) G B(F).
We blow up the closed point corresponding to x on Vr ® к. Let D be its in-
verse image on the resulting surface V". The composite morphism
V" -+ V' ® к V ® к is in 5(K ® kf and (^„(D) defines an element of
the group 7V(K”), which is canonically imbedded in Z ’ (F). We let the image
of the class of Oy(D) under this embedding correspond to the point x.
Briefly: each point x E E(/)(M is put into correspondence with the class of
its blown up bubble; this map is extended to Z°(K) by linearity. It is easy to
see that the map is well defined.
The main justification for introducing the space E (K) is the following:
Proposition 35.6. The homomorphism just constructed,
N(V)X.Z°(V')->Z'(V),
is an isomorphism of G-modules with scalar product.
The proof is obtained by the completely formal passing to the limit from
Lemmas 34.1 and 34.2. Roughly speaking, ‘on a finite level’ of one monoidal
transformation /: V’ -> V with as centre the point x (over k\ we write
Pic V' = Pic V Ф Zx ,
where (x, Pic V) - 0 by 34.1 (iii), Pic V denotes/* (Pic V) and x stands for
the class of Oy, (/-1 (*)).
35.7. In future we shall identify Z’(K) withTV(F) X Z°(K) as de-
scribed in 35.5. The following intuitive interpretation suits this formalism.
Let I E7V+(K). Then I is uniquely determined by the complete linear sys-
tems of the zeros of the sections of the corresponding invertible sheaf.
Now let x E F(fc) be some point. We consider the element I — x E
N(V) X Z°(F) - Z * (K). Let f: V' -> V be the blowing up of x. Then l~x repre-
sents some element T in TV(L'). We suppose that l' &N+(V’) as well and con-
struct the linear system of the sections of the corresponding sheaf on Vf and
its image under f in V. Then this will consist of precisely those sections of the
sheaf of I which pass through the point x.
Similarly, the element I - Ъа^ corresponds to those curves of the linear
system of I which pass through the point x^ with multiplicity > a p through
the point x2 with multiplicity > л2, etc. (cf. Lemma 35.10 below).
Needless to say that the inconvenience of this interpretation is connected
with the existence of ‘virtual linear systems with prescribed base points’,
which are not linear systems at all. The group N(V) X Z®(V) provides also a
convenient framework for such objects. The interpretation connected with
effectiveness is reflected in the structures of Z+* and Z± .
Finally, we have started with the introduction of Z * (7) because the defini-
tion of this group is more invariant and more convenient for the verification
of functorial properties. On the other hand, the group 7V(F) X Z°(K) is well
suited for concrete calculations, as we shall see below.
Finally, we shall establish some auxiliary facts on the extra structures in
Z-(F).
Lemma 35.8. Let L be an invertible sheaf on V ® к which is isomorphic to
/*(Opr(l)), where f : V ®k ~>Pr issome morphism. Then the class I of this
sheaf in Z* (K) belongs to Z^\V),
Proof. Obvious.
Lemma 35.9. The restriction to Z°(V) of the homomorphism
^2 :N(V)X ZQ(V) Z (see Section 34.7 (v)) is defined by the formulae
Q(x) = — 1 for all x G E(F) (k).
Proof. In fact, for every exceptional class / € Pic (Vr ® k) we have
— 1.
Lemma 35.10. LetD С V® к be some effective divisor, let x^,... xn^D
be closed points, and let n^ be the multiplicity of X} on D. We denote by
I EN(V) the class of the sheaf Oy(D) and identify the points with the cor-
responding points of E(V)(k). Then
n
i -S niXi&z;(v)..
i=l
Proof. Let/: V' -+ V® к be the monoidal transformation with as centre
the union of the points Xy According to Proposition 20.4, the class
represents in the group Pic V1 the class of (£>)), where /"ЦР) is the
proper inverse image of D (Definition 20.3). As/-1(D) is effective, everything
is clear.
Corollary 35.11. Letx, x" EE(V)(k).If x <x", then x" —x EZ^(V).
Proof. Let x' lie on a surface V-, and x" on V". If the birational map
Vf -> V" is locally the blowing up of x", then the element x” — x' represents
the element l x’ (in the group 77(7') X Z®(V') = Z\V') = Z'(Vyf where
I is the class of 0y,(D) and D is the inverse image of x" on Vf. In this case,
the assertion follows from Lemma 35.10 because x lies on D.
In the general case, there exists a sequence of points x" ^Xj > . .. >хц~х
such that (%p x/+1) satisfy the preceding condition. Therefore X/-x?-+1 € Z^(V),
and hence x" — x E Z*(F) because Z\ is a semigroup.
Corollary 35.12. Let I - Snz-xz- GZ+’(K), where lEN(V) and x2- E E(V)(kf
Then'.
(i)ZGW+(F).
(ii) nt > 0.
(iii) nt > П] if Xf >Xj,
Proof. The element I - Zn^Xf represents some class Г EN(Vf) on a surface
V' 4 V. It is obvious from the definition that even V EN*(V'f But then
Z = /*(Z') G7V+(F') because of 34.3 (vi).
Further, Xf EZ±(V) (obviously) and Xf -Xj EZ^(V) if Xf >Xj (Corollary
35.11). Therefore
nf = (Z- 2 n.x., Xf)>0 ,
ni ~ nj = ~ 2 nixi> xrxj) 0 ’
which concludes the proof.
Lemma 35.13. Let f: V' -> V be some birational map. We suppose that
there exists an element V EN(y’}E Z'(V') such that I ~ f(l')EN(V) E
CZ‘(V) and I is the class of some ample invertible sheaf on V. Then f is a
morphism (and hence Z' =f*(l)).
Proof. Represent /as a composed map V' F, where g and h are
birational morphisms. We want to prove that h о g-1 is a morphism, i.e., that
h collapses all curves which g-1 blows up.
An easy induction on the index of g permits us to restrict ourselves to the
case where g is a monoidal transformation with a closed point x G V’ as
centre. Let GvV(F1) be the class of the inverse image g-1 (x). Applying
Lemma 34.1 (ii), (iii), we find
o = UV), zp = = (/, Mi)).
It follows from this that h collapses the curve g-1(x): the opposite case the
class of h*(ly) ^N(V) would represent an effective divisor, and the intersec-
tion number of the ample class I with any non-zero effective divisor is positive.
This proves the lemma.
36. Calculations on cubic surfaces
We now apply the developed techniques to cubic surfaces. We start with
the calculation of the actions of the birational maps tx and sxy: V -> V on
the group Z’ (F) (see Theorem 34.8).
Lemma 36.1. Let V be a cubic surface over an algebraically closed field k,
let x у (resp. x^Xq) be a good point (resp. a good pair of points) on V (see
33.6). Then for any element
к
z=au>v+T)b.x. GJV(F) X Z°(F) = Z’(F),
i=l
where x.€E(F) (k) and x. Ф x- for i =#/, we have
Z I J
к
rx1(z) = (2a-z’i)wK + (3a-2Z’l^l + 'Lbitx (Xj), (V.5)
z-2
y (z) = (5a—2b.—2b^)<joJ7 + (6a — 3b^—2b^)x.
ЛРЛ2 -I V 4 1 z7 1
К
+ (6a-2b1-3b2)x2 +DblsXi (x?) . (V.6)
i=3
Proof. We start with the explanation of the symbols tx (xf) and sx x (xz«).
Let V' -> V be the blowing up of the point Xj. Then E = E^Xfc^fX]}.
But tx is a (biregular) automorphism on V': see Example 21.2. Therefore
tx can be naturally extended to an automorphism of the space E(K')(A:).
We denote the image of xf- under this action by tx} (xz) for xz Фх j.
Similarly, we can check that sx x^ = Lc ^xt ox2 acts biregularly on the
surface V" obtained from И by blowing up the points Xp x2 (but notX|OX2!).
This action can thereupon be extended to E(K")(£) = E(F)(fc) ^{xp
which permits us to define the points x2(xz) f°r z 1 >2.
We now prove formula (V.5).
First step. Let/: V' V be the blowing up ofxp We identify by means
of this morphismZ‘(F) = 7V(K) X Z°(F) withZ’(r') = 2V(K') X Z°(K') so
that E(L')(/:) = E(K)(A:) \{xj] as above. Denoting by GTV(K') the class
of the inverse image of Xj, we then have
coF = copX1=Z, x^x^O^l),
so that
z = acoy, + (b-a)I + Z/ bixi. (У-Ъ
i=2
Second step. For brevity let us denote by tx the automorphism of V’ in-
duced by tx on V. We calculate its action on tAe (for us interesting) elements
of Z’(K').There is nothing to say about the action onxf; moreover, tx^(cov,)=
— COyr .
We now show that tx (Z) = -coy, ~1. Indeed, let D С V be the intersection
of V with the tangent plane in x. Then x has multiplicity two on D, so that
class/*(D) = class/-1(D) + 2 /.
But
class/*(D) = ~/*(^>r) = + h
so that
• class/-1 (Z>) + / = — co у, , I- class / 1 (x) .
On the other hand, the geometric analysis (see Example 21.2) shows that the
curves and/^Cx) are interchanged under the action of tx . This
proves what we want.
Third step. Substituting tx (/) = -coy, --1 in (V.3), we obtain
к
*x. CO = (2л - b} co y. + (a - b) I + Z/ b. tx (x.) .
1 r=2 1
Finally, under the opposite identification ofTV(F') X Z°(K') with
JV(P) X Z°(F), the class coj/' goes to cov + xb the class I to xb and tx^)
remains the same. This proves (V.5).
Formula (V.6) is obtained by a mechanical threefold application of formula
(V.5), if one takes into account that sv v = tY tY tY . We omit this calcu-
v 75 ХЪХЪ *1 -y1ox2 Л2
lation; note, however, that precisely here the hypothesis is used that we
have a good pair (Xj,x2) and not just two separate good points Xp x$.
36.2. Comments. From formula (V.5) it follows that
wjz) 2cOpr —3X| .
According to Section 35.7, such an equality means that the linear system of
hyperplane sections of V goes into a linear system of intersections of V with
quadrics which have at least a threefold point inxp If one prefers, the content
of the lemma is exhausted by this.
This classical terminology is completely adequate as long as points lying
over V do not turn up. Within the frame of our formulation, such geometric
difficulties are easily localized and separated from the purely algebraic theory
of the representation of birational maps inZ* (K).
37. Birational non-triviality
It is time to start gathering the fruits.
We start with the proof of Theorem 33.1, which is obtained as a corollary
of a much stronger result. To be able to formulate this exactly, we call a k-
surface V a surface with a rational bundle structure if it admits a fc-morphism
f\ V В such that the base and the general fibre have genus zero (the fibre is
supposed to be irreducible and reduced).
Among these surfaces is for instance P1 X P1, so that the birationally trivial
surfaces belong to this class. It is a very large class; for instance, if A; is a finite
field, then it contains representatives of all except a finite number of the
classes of surfaces birationally trivial over к modulo birational equivalence
over k. This is proved in Manin [4].
Theorem 37.1. A minimal cubic surface V over a perfect field к cannot be
birationally equivalent with any surface with a rational bundle structure.
Proof. Let К be birationally equivalent with such a surface. Then the image
z in Z’ (F) of the class of some fibre of the bundle has the following proper-
ties:
zez+(nG, (*,*)= o, n(z) = -2.
(V.8)
In fact, all this is true in the Picard group of a surface with bundle structure
and it is preserved in Z* (K) according to Theorem 34.8. We shall show that
there are no such elements.
According to the minimality criterion 28.1, N(V)G ~ Zcoy, Therefore z
must have the form
a
z = —аЫу — Zv bixi.
Corollary 35.12 then shows that a > 0 and Щ > 0 for all i.
Choose a z with the lowest possible value of a. By Proposition 35.6 and
Lemma 35.9, the last equalities of (V.8) can be written in the form
к
За20,
Z=1
к
За - У/ bj = 2 .
Z=1
(V.9)
We can assume that Zjj > d2 > 0 (it is clear that not all Z>z- can be
zero). According to Corollary 35.12 (iii), in any case one of the points of
maximal multiplicity b^ must lie on the surface V and not over it. Let this point
bejtq F(£). We show that it is a good point. Otherwise it lies on some line
belonging to an exceptional class Z EtV(K). According to Lemma 35.10,
I ~~x± € Z^ (K), from which
0 < (Z-Xp z) -a-by .
This, however, contradicts eqs. (V.9) because
К К , К .
3<z2 = Zz b? < by 23 < by I 23 bt + 2| = 3aby ,
i-1 z=l u~l f
which implies a < by. Because is a good point, we can apply the automor-
phism tx to V0 k and calculate its action onZ’(K) by means of Lemma
36.1. Then
(v.(X) Q# (3# 2by)Xy (x.).
1 1=2 1
Moreover, by Theorem 34.8 the element tx (z) also satisfies conditions (V.8)
(just as z). But the coefficient of — w у of it is equal to 2a - by < a, and this
contradicts the choice ofz. This proves the theorem.
38. Birational classification
In this section Theorems 33.2 and 33-.7 will be proved. They are both con-
tained in the following result:
Theorem 38.1. Let Vf Vf be cubic surfaces over some perfect field k, let
V be minimal and let f: V' V be some birational k-map. Then there exists a
birational k-map g: V -> V with the following properties:
(i ) g can be represented as a product of maps tx and sy^z for good points
x E V(k), respectively good pairs of points у, z E V(k) defined and conjugate
over a quadratic extension of the field k.
(ii ) The composed map И'-Д Vis an isomorphism.
Corollary 38.1.1. Under the conditions of Theorem 38.1, V* is also minimal.
38. 2. The derivation of Theorems 33.2 and 33.7. For Theorem 33.2 it suf-
fices to make sure that every isomorphism h : V' Vis projective; but
Л*(Ор*) = Пр*, and Пр1 is precisely the sheaf 0y( 1) under the standard em-
bedding.
Theorem 33.7 is obtained by applying Theorem 38.1 to the case V' = V
and f an arbitrary birational map of V into itself.
38. 3. Proof of Theorem 38.1. We set z = /(—<Op,) €Z‘(F). This element
possesses the following properties:
z6Z+'(K)G; (z, z) = Q(z) = 3 .
Setting z = ~~aa> у ~~ S?=1 b.x., as in Section 37, we find a > 0, > 0 for all i
(we again use the minimality of V in the form of the equality N(V)G - Zcop).
Further,
к
За2 — S b2 = 3 ,
i=l
(V.10)
к
3a-Z/bz = 3;
z-1
therefore a > 0. We suppose that not all the are equal to zero and set
Z>1 > > . .. > bK > 0. We have
3a2 = Z)z>? + 3<Z>.( S&.+ 3
Z=1 \ z-1
consequently a <b^. (The equality sign is possible here only if b^ - 1, but
then a = 1, which contradicts (V.10).) It follows from this that there can be
no more than two points with the coefficient by otherwise За - <
< За — 3b^ < 0.
We now consider two cases separately; and these exhaust all possibilities.
Case 1. On the surface Vlies exactly one point of Xy..., xK with coeffi-
cient by Let this point be Xy Exactly the same arguments as in Section 37
then show that x1 is good. Moreover, the element z is G-invariant, therefore
Xj G F(£). Applying Lemma 36.1, we find from this,
=3ab^ ;
к
tx (г) = -(2<z-bjuy -(3a-2b])x1 -Tjbitx (x) ,
1 z=2 1
so that the coefficient of — co у has diminished.
Case 2. On the surface V lie exactly two points of Xj,.. . , xK with coef-
ficient by. Let these bejq, x2. The pair (x^x2) is G-invariant because z is G-
invariant. Consequently, either both points are defined over k, or they are de-
fined and conjugate over a quadratic extension of k. In the first case it suffices
to apply tx to diminish the coefficient of a. In the second case we can make
use of sx x (see Lemma 36.1), but we must first show that (xp x2) is a
good pair.
Because both of the points Xp x2 (having maximal multiplicity) are good,
difficulties can only arise if the line through Xp x2 is tangent to V at precisely
one of these points; but this is not possible because such a line is G-invariant..
The other bad case is when the point Xj о x2 lies on one of the lines. We
show that this is not possible. In fact, according to Lemma 36.1, the coeffi-
cient of —coу in the decomposition of the element (z) is equal to 2л - Z?2,
and the coefficient for Xj о x2 is equal to Z>p But 2a - ~ 2a-by < by;
therefore Xj о x2 is a point whose multiplicity in tx (z) is greater than that of
— coy. The same arguments as in Section 37 applied to tx (z) instead of z
show that Xj о x2 is a good point.
Conclusion of the proof Repeating the above constructions, we construct
a map g: V V by composing the maps tx and sx y such that the composed
map go/: E'V-+ Khas the property g о f(—co y,) = —acoy. Applying
Lemma 35.13, we immediately obtain that g о f is a morphism. But it can
collapse nothing because rkA/K') = rkAr(K) = 7. Therefore go /is an isomor-
phism. This concludes the proof.
39. Relations between the generators
In this section, Theorem 33.8 will be proved, as a corollary of some more
general results.
Excepting the last two subsections, the base field k is supposed to be alge-
braically closed in this section; V is an arbitrary smooth cubic surface over k.
Definition 39.1. A birational map f: K-* V over к is called good if under
its action on Z '(F) the subgroup Zgo^ + Z°(K) is mapped into itself.
Motivation: If/comes from a birational map of a minimal surface over some
subfield of k, then/is good; this follows from Theorem 33.7 and Lemma 36.1.
We denote by В the group of good birational maps of V. The coefficients
of its representation in + Z°(K) will be our main tool in the study of this
group. We therefore give the following formal definition.
Definition 39.2. The functions a,bx‘.B Z (where x G E(V) (k)) are given
by the formula
/'(w7)=a(/)wF + Z/ix(/’)x. (V.ll)
For convenience of reference, we restate in this slightly more precise con-
text a series of results which have already been proved.
Lemma 39.3. For every element fEzB we have
Jx(/)>0, bxO)>b(O
✓V -A- J »
ifx >y.
This follows from Corollary 35.12.
Lemma 39.4. Let x be a good point of V(k), and let (x,y) be a good pair.
Then tx &B, sxy EB, and forallf^B, we have
a(txf')~2a{f)~~bx{f)i
bxexf) = 3a(f)-2bx(f), (V.12)
ьхоУехл=ьу{п
and
a(s vf) = 5aO)-2bx(f)-2b (f), (V.13a)
-A j У tAr
bxWvf)=6a(O-3bx(O-2b (f), (V.13b)
by(sx yf) = 6a(f)-2bx(f)-3b. (f), (V.13c)
J' 3 у J'
\y{z^sx,yf^b^ ifz*x,y. (V.13d)
This is an obvious reformulation of Lemma 36.1.
Lemina 39.5. Let fEB.
(i) If a (/) > 1, then there exists a good point x 6 E(K) (k) lying on V
such that bx(f) > a(J). No more than two points of E (K) (k) satisfy this in-
equality.
(ii) Let x,y E E (7) (k) be two good points lying on V such that bx(f) >
> by(f) ^a(f). If at least one of these inequalities is strict, then they form a
good pair.
(iii) Let x G E (7) (k) be a good point lying on V. Then bx(f) Ф «(/); in
particular, in the conditions o/* (ii) one of the inequalities is certainly strict.
Proof, (i). We have
/(w jz)) ~ ~ 3 >
^(/(^jz))= = (<0^,60^) ~ 3 >
from which
3a(f)2 - E bx(f)2 = 3 ,
X (V.14)
за(П-ЕАСО = з.
After this, the argument goes on as in the beginning of the proof of Theorem
38.1.
(ii) . Let (x,x) G E(7) (k) be two good points lying on V and such that
£x(/) where at least one of the two inequalities is strict. To
prove that (x, _y) is a good pair, we must check that the line through x,y is
not tangent to V in x,y and that x о у does not lie on the lines of V.
Suppose that this line is tangent to V at x (the case of у goes completely
analogously). The tangent plane to V at x intersects V in a curve which passes
through у and which has a double point at x. This curve belongs to the class
of — coy. According to Lemma 35.10, — co у — lx —y 6Zl(V). Because
6Z+'(n>we obtain from this
_ 2x-yj(cov)) - 3a(f) ~ 2bx(f) - by(f) > 0 ,
but this contradicts the condition imposed on x, y.
We now show that x о is a good point. In fact, by Lemma 39.4
a(t f) = 2a(f) - b (/) < b (f) = b it ft.
X X л X л X
If х oy lies on a line of class /, then / — x о у GZ'+(V) by Lemma 35.10. On
the other hand, tyf(— coy) GZ+'(K). Therefore
0<(l-x°y, t f(-w„)) = a(t f)-byOJC(t f) .
The contradiction thus obtained proves what we want.
(iii) . Suppose that there exists a good birational map/€2? and a good
point x G V(k) such that bx(f) = #(/)- We choose an/with a smallest possible
zz(/). It is clear that a(f) > 1; otherwise equation (V.14) could not be valid.
Let у G V(k) be a good point such that by(f) > a{ff it exists in virtue of
assertion (i). According to property (ii), (x,y) forms a good pair. From
Lemma 39.4, eq. (V.13), it follows that
а^хуП = Ъа(П-2Ьу(П,
bx(sxvf) =3a(f)-2b (f).
л лгХ X
But this contradicts the choice of /because sx ^/is a good map and
3a(f) — 2b (f) < a(f). This concludes the proof.
Lemma 39.6. a(f) - 1 if and only iffis a projective automorphism of V,
Proof. From (V.14) it is clear that all bx(/) = 0 if a(/)= 1. Applying
Lemma 35.13 to the map/: V -> V we obtain that /is a morphism. We have
already verified that in this case/is a projective isomorphism.
The last result parallels Theorem 33.7.
Theorem 39.7. The group В of good birational maps of V is generated by
the group W of projective automorphisms and the maps tx for all good points
x.
Proof. Suppose that this is not the case. We consider an/GT? with smallest
possible й(/), which is not contained in the subgroup generated by W and the
It is clear from Lemma 39.6 that a(f) > 1. According to Lemma 39.5 (i),
there exists a good point у G V(k) for which by(f) > a(f). Then tyf G В and,
by Lemma 39.4,
a(tyf) = 2a(f)-by(J) <a(f).
This contradicts the choice of /because ty f does not belong to the group
generated by W and the tx if/does not. This proves the theorem.
We shall now occupy ourselves with the relations. The basic result of this
section is:
Theorem 39.8. The following relations between the generators w G W and
tx of the group В generate a full system of relations:
t t t = t t t ,
x хоу у у xoy X ’
Wtxw~1=twM’ <v-15)
r2 = l.
In particular, В is a semidirect product of W and the normal subgroup В gener-
ated by all the tx.
39.8.1. Start of the proof That the first and the third of the relations (V.15)
hold has been proved earlier. The second easily follows from the fact that the
action of w is linear; therefore it can be extended to the whole space P3 and it
sends the lines by means of which tx is defined into lines. In particular, good
points and good pairs remain good.
We now occupy ourselves with the completeness of the system (V.15).
We denote by D the free product of the groups Z2 generated by the symbols
Tx and Sx,y> one for each good point x G U(&) and one for each unordered
good pair of points x, у £ K(fc), with the relations = S^. y = 1. Construct
the semidirect product DW with the normal subgroup/) on which W acts ac-
cording to the rule
wTxw~X=Twtx)’ wSx,yw~X =Sw(x),wW
There exists an epimorphism D W В which is the identity on PE and which
sends Tv into and into ty, (by definition!).
To prove the theorem it suffices to verify that the kernel of the homomor-
phism D W В (as a normal subgroup) is generated by the elements
Sxу Tx TXQy Ty. First we describe how to associate to each element of/? some
uniquely determined inverse image of it in DW.
Each element unequal to the identity ofZW can be uniquely written as
R^2... Rn, where Rn is one of the symbols w G W \ {1}, Tx, SXy, and R^
for z < я— 1 is one of the symbols Tx, Sxv. Moreover, R^ Ф 7?z-+1 for all i.
We shall say that this word starts with R ; the number n is called its length.
The identity is represented by the empty word of length zero. We shall identify
the elements of DW with such words.
We define canonical words by induction on the length. The definition uses
in an essential way the representation of DW on Z '(7) induced by the homo-
morphism DW B.
Definition 39.8.2. (i) The empty word and all words of length one are ca-
nonical.
(ii) The word TXF is canonical if and only if the word F is canonical and
satisfies the conditions
bv^txf^<bx^txfy> for all у lying on И,
where / is the image of F in В.
(iii) The word 5Y VF is canonical if and only ifF is canonical and satisfies
the conditions
>
b (s f)~b(s f}'>a(s f).
x,yJ 7 yv x,yJJ v x,yJJ
The continuation of the proof of Theorem 39.8 has been split into several
lemmas.
Lemma 39.83. For every elementgEB there exists a unique canonical
representative G in DW.
Proof. The existence of G is established by induction on the number «(g).
If «(g) = 1, theng G W according to Lemma 39.6, so that the element G is
canonical. Let «(g) > 1 and suppose that it has already been proved that there
exists a canonical representative for every fGB with «(/) <«(g). Find a
good point x E V(k) such that bx(g) >«(g) (Lemma 39.5 (i)). If in addition,
bx(g) >by(g) for every other pointy on V, we set f- txg. Then «(/) =
= 2a(g) - bx(g) < «(g). Let F be a canonical representative of/. Then TXF
is a canonical representative ofg by Definition 39.8.2.
If there exist precisely two points x, у on V such that bx{g) ~ by{g) > «(g),
they form a good pair by Lemma 39.5 (ii). We then set f~sxyg. Applying
(V.13), we find «(/) < «(g). Denote by F a canonical representative of f;
then Sx yF is a canonical representative ofg.
Finally, the uniqueness of the canonical representatives follows from the
fact that the element it starts with is uniquely defined according to Lemma
39.5.
Example 39.8.4. If x, у € V(k) form a good pair, then (Ty Tx)n is a canon-
ical representative of (ty tx)n for all n > 1. In particular, the element ty tx
has infinite order.
Proof. We set fn = (ty tx)n. We first of all check that bz{fjf) ~ b2(tx fn) = 0
if z Ф x, y, x о у. This is true for n = 0, and it is easily established by induction
on n for the remaining n by means of Lemma 36.1.
The proof that (Ty Tx)n is canonical also goes by induction.
Induction hypothesis for n\
bxoy(fm^ < ’
2аЮ>ЬхЮ + ЬуЮ fora11
(For n = 1 we have «(Д) = 4, hx(/j) = 0, by(J\) = 6, bXQy(J}) = 3, according
to Lemma 36.1, so that all inequalities are true.)
Corollary for n: the words (TyTx)m and Tx(TyTx)m are canonical for
all 1 < m <
In fact, it is clear from the definition and the induction hypothesis that the
canonical representative of (tytx)m starts with Ty. Further, since
^(44,)= 2a<4P - bx(fm) .
Ьх^хЫ = 3<Ы-2Ьхит)’
^xoy^x ’
by ) ~~ &хоу ’
we have
^xoy^x by^xU ’
so that the canonical representative of tx(ty tx)m starts with Tx. Clearly, our
assertion follows from this.
The step from n to л +1. We have, using Lemma 36.1 twice:
а(Г„+1) = 4a(/„) - 2bx(fn) - Йхо/Г„) ,
by (4+i) = 6a(/n) - 3 W - 2ЬхОуЮ >
WW'W'W
All inequalities of the induction hypothesis for n +1 can now be checked mechan-
ically. We leave the details to the reader.
Lemma 39.8.5. Let fE В and let x, у be points on V such that bx(f)>
> by (/) > a(f). Then the canonical representative off has the form Tx Tx Qy G,
where G is a canonical word.
Yxwi. It is clear from the definition and Lemma 39.5 (i) that the canonical
representative of/starts with Tx. Therefore it suffices to verify that the ca-
nonical representative of tx f starts with TXQy.
It follows from Lemma 39.5 that the pair (*,y) is good; therefore x о у is
a good point. Moreover,
*xoy( W > 2й(/) - ^(/)= “(txn *
We check that bz(txf) < bxoy(txf) for all z Fx о у. If z =x, then
\(ГХГ) = 3a(/) - 2bx(f) < b(f) = b (txf) .
«Л- Л _A У *A У «А
And if z ±x, then z = x о и (и Фу,х), and then the inequality we need,
•A4^ 1Л Л M у
follows from Lemma 39.5. This concludes the proof.
We now prove the fundamental lemma from which Theorem 39.8 can be de-
duced in a few words.
Lemma 39.8.6. The canonical representative of a product of elements of
В can be obtained from the product of the corresponding canonical represen-
tatives by applying the relations (V.15) (more precisely, their analogues in
the group DW) and the relation ,, = Tv v T.,.
Proof. It suffices to consider the product of two elements. Induction with
respect to the length of the canonical representative of the first of them im-
mediately reduces the matter to the case where it is equal to w, Tx ovSx y.
Let F be some word in D W. We denote by Fw (w E W) the word which is
obtained from F by conjugation:
Tx TW(x)’ Sx,y * Sw(x),w(yy W1 •
Obviously wF = Fww. Because the action of H7 on V is linear, F is canonical
if and only if Fw is canonical. This proves the lemma in the case where the
first factor is w.
If the first factor is equal to Sv v, applying the relation .. = T, TY n., TY
we reduce the matter to the case of a first factor Tx. Only this case presents
difficulties.
Let F be a canonical representative of the second factor and let/be its
image ini?. We must examine the canonical representative of the element
fx/and compare it with TXF
The case a(f) ~ 1 is trivial. Suppose that a(f) > 1 and that the assertion
of the lemma for tyg has already been proved for all g with я(g) < and
for all points у E V(k}. The word TXF itself is canonical if the following con-
ditions are fulfilled:
bx(txf) > а^х^> >
ЬУХГ) > bxov^x^ for a11 x ° У on V
(Herex о у = rx(y) by definition.) Non-trivial possibilities arise only if these
conditions are not fulfilled. This can only happen in one of the following two
cases:
(i) bx(txf) < a(txf). The equality bx(txf) = a(txf) is impossible by Lemma
39.5 (iii).
(ii) . bx(txf) > there is a point x о у on V such that bx(txf) <
^bxoy(.txf\
We study these two cases separately.
Case (i). We have
bx(txf) = 3a(f) - 2b (f) < a(tf) = 2a(f) - bx(f) ,
from which bx(f) It is evident from the definition that the canonical
representative/7 of the element /either starts with Tx or with 5^ for some
pointy. If F = TXG. then G is a canonical representative for txf, which is ob-
tained from the product TX(TXF) by means of the relation Tx - 1.
We now suppose that F ~ Sx y G. We show that in this case the canonical
representative of tx f is equal to Txoy Ty G. This word is obtained from the
product TXSX y G by applying the relation Sxy -Tx Txoy Ty.
By definition, Sx yG is canonical if and only if the inequality bx(f) =
= by(f) > «( /) holds. Consequently, the pair (x, y) is good and the point
x о у is good. Therefore,
bxov(txf) = W > Я(Г) > W) - bx(f) = a{tf) > bx(txD .
Moreover, bXGZ(txf) < bxoy(tx f) for all z £x, у by Lemma 39.5, which
shows that bz(f) < by (/) for such z. It follows from this that the canonical
representative of txf starts with TXGy.
It remains to show that the canonical representative of txoy /starts with
Ty. One needs to check that
(V.16)
by(jxoy fx^ > bz^xoy fx Г) > for 311 Points z^y tying on V
The first inequality of (V.16) follows from the relations
^хо/х^ = Ьхо(хоуУхоу*хП = Wx^ = W) - bx(f) ,
a(fXOV = 2а^хП - Ьхоу((х^ = - 2bx^ - W ’
because by(f) >«(/). The second inequality of (V.16) must be verified sep-
arately for z = x о у and for all the remaining points z Ф x, у, x о у, each of
which can be represented in the form z = txoy tx(u).
If z =x oy, we have
Ьхоу(1хоУхП = 3a(txf) - 2bxoy(_txf) = 6a(f) - 3bx(f) - 2by(f)
</’Ло/хП = 3Д(Г)-2^(/)>
because b (f) = b (/) > д(/) .
Л Jr
If z = t tx(u), u=£ x, у, XO у , then
bz{t fx/) = bu^ < - Ьх(П = ЬУхоу‘хП ’
because in the opposite case we would have
bx(J) + by(J) + bz{f) > 3a(f) ,
which contradicts eqs. (V.14).
Case (ii). First of all, in this case
\(/) <«(/),
by(^ = bxoy(fxf)> bx^x^ = 3й<Г) - 2M/) > й<ГхГ) > а(Г) •
Consequently, the pair (y, x oy) is good and the pointy is good. We show that
bz(f)<by(f) ifz^y. Indeed,if bz{f)>by{f), thenz#=x and
W = ^oz(V)>Z’xoyV)
from which
bxoZ 0xn + b (txn + bx(txf) > 3a(t f),
which contradicts (V.14). It follows from this that the canonical representa-
tive of f starts with Ty. It remains to check that the canonical representative
of tyf starts with Txoy .For this one needs to establish two inequalities:
(V.17)
bxov(t f) > b (t f) for all points z =# x о у.
We have
ЗДЛ = bx(f) , a(tyf) = 2a(/) - by(J) ,
and
bx(f) > Kn - kbytf) > 2a(f) - by(f) ,
from which the first inequality of (V.17) follows. The second inequality will
be verified separately for z = y and for all other points z фу, x°y, each of
which can be represented in the form z = ty(u). If z -y, we have
ЬхоУуП = bx(f) > by(tyf) = 3a(f) - 2by(f) ,
because
bx(f) + 2bv(f) > 2bx(f) + b m > 3a(J) .
If z =y о и, then from the inequality
byoyyn>bxoy(tyr)
it would follow that bx (/) < bu(f), and therefore
МП + b(f) + &„(/) > bx(J) + (За(Г) - 2ЙХ(/)) + bu(J) > 3a(f),
which contradicts (V.14). Thus in case (ii) the canonical representative of f
has the form F~Ty TxoyG, where G is a canonical word.
If bxoy(txf) = bx(txf\ then the canonical representative of ^/starts with
Sx xoy>ап^ hence it is equal to Sx, XQy G, i.e., it can be obtained from
the product Tx Ty TxoyG by means of a standard relation. Finally, suppose
that bxsyy (txf) > bx(txf). Then according to Lemma 39.8.5 the canonical
representative of tx / has the form Txoy TyG', where Gr is the canonical repre-
sentative of txg (g is the image of G). But the word Txoy Ty Tx G is obtained
from the product Tx(Ty TXQyG) by applying the standard relations Tx TyTxoy-
= 'S'v Ymn vnv “ l, and the word G’ is obtained from the word TYG by
means of the relations specified in the lemma in virtue of the induction hy-
pothesis because a(g) < a(f). (This is the only place where induction is used.)
This completes the proof of the lemma.
39.8.7. End of the proof of Theorem 39.8. Let the element F&DW belong
to the kernel of the homomorphism DIV By Lemma 39.8.6 this means
that by applying to F the relations in the group DIV and the relation
Sxy~Tx TXGy Ty we can obtain the empty word. This completes the proof.
39.9. Application to minimal surfaces. Starting at this point, к denotes a
perfect field, У is a minimal cubic surface over k, and G = Gal (к /к). We de-
note by H some group of projective ^-automorphisms of V. We set Gy = G X H.
This group acts in the obvious way on 7V(K) and on Z*(K). The following re-
sult contains Theorem 33.7 as the particular case where H = {1}.
Theorem 39.10. We suppose that N(V)G1 = Zeey. Then the group ofk-bira-
tional maps of V which commute with the action of Gy is generated by the
following maps'.
(i) The centralizer of H in Aut^(K) С Ж
(ii) The maps tx for H-invariant good points of K(k).
(iii) The maps sxyfor G у-invariant good pairs x, у G V(k).
The relations between these generators are those described above.
Proof. Extending the action of G* to E (V) (k) in the obvious way we ob-
tain for all g G Gy:
~^{х}> 8sx,y£ ~ Sg(x)>g(y) ‘
We apply these relations to a canonical representative of a к-map which com-
mutes with Gj. It remains canonical. Using its uniqueness, we obtain that all
components of length 1 which go into this representative must be Gj-invariant.
From this the assertion on the generators follows. A full system of relations
is obtained precisely as in the preceding theorem.
40. Bibliographical remarks
This chapter is completely based on material from Manin [3] and [8].
Theorem 33.1 was first proved by Segre [3]; an analysis of its proof, which
has been reproduced in other terms in Section 37, also led to a series of further
specifications. The arguments of Segre in turn go back to Max Noether, who
was the first to describe the generators of the group Bir P2 over an algebraically
closed field. Note that the relations between the generators in the theorem
were unknown to Noether.
As the group Bir V is so closely connected with the Appoints of a minimal
cubic surface К one may hope that this very group reflects the essential arith-
metical properties of V over local or global fields.
One should probably start with a study of the various topologies on Bir V
which are induced by the local topologies of the field. Does there exist a unique
invariant measure?
The group of projective automorphisms W is trivial for almost all surfaces
V (i.e. on a Zariski open subset of the space of coefficients). Some account
of the ‘exceptional’ V for which W is not trivial can be found in Segre [1].
CHAPTER VI
THE BRAUER-GROTHENDIECK GROUP
41. A survey of the results. Obstructions to the Hasse principle
41.1. We first recall the basic information on the Brauer group of an arbi-
trary field к (see Bourbaki [2]; Serre [3, 6]; Cebotarev [1]).
A finite-dimensional fc-algebra A is called a simple central algebra over к if
there exists an n > 1 such that A ®k к ~Mn(k) where к is the algebraic clo-
sure of к and where Mn denotes the algebra of {n, ri) matrices.
The tensor product induces on the set (of classes up to isomorphism) of
central simple algebras over к the structure of a commutative semigroup.
The following equivalence relation turns it into a group '. An algebra A is
equivalent to В if there exist numbers m, n such that А ®kMn(k) is isomor-
phic to В ®kMm(k). All the matrix algebras over к are equivalent to one an-
other and they form the zero class. The class of the algebra A °, the opposite
algebra to A (that is, the algebra with the same elements and the same addition,
but with the multiplication performed in the inverse order) is the inverse to
the class of A. In fact, the canonical map A ®k A® End^(4 ) (the endomor-
phisms of the linear space A) which associates multiplication by x on the left
and by у on the right to the element x ® у E A A0 is an isomorphism: its
kernel is trivial because A ®k A0 is simple, and the dimension of A ®k A0
is the same as the dimension of End^ A, i.e., (dim^ A)2.
The group of classes of central simple algebras over к up to equivalence is
called the Brauer group of the field к and it is denoted Br k. It admits the fol-
lowing cohomological description.
Let К D к be some extension of the field k. It is called a splitting field of the
^-algebra A if A ®kK = Mn(K). Equivalent algebras have the same splitting
fields. Let Br (к, K) be the subset of the Brauer group consisting of the classes
of algebras with splitting field K. It turns out to be a subgroup.
Suppose now that К D к is a Galois extension with Galois group G. Then
one can establish the following fundamental isomorphism:
Bi(k,K)~H2(G, A*).
It admits various descriptions. We here indicate one of them which is needed
for the following: the so-called ‘crossed product’ construction. It consists of
an explicit construction of a central simple algebra over к given a ‘factor system’,
that is, a cocycle {atf} EZ2(G, K*). This algebra A is constructed as follows:
A = © Kes ;
s&G
еЛ = аЛ’ VV6G’
e a ~s(a)e Vs EG, a EK.
& о
Its dimension over к is clearly equal to [K : k]2. We omit the verification of
all the properties needed for the construction; we only remark that the asso-
ciativity of A is equivalent to the fact that the cochain of‘structure constants’
{zzs t] is in fact a cocycle.
We now introduce the basis for the subsequent algebraic-geometric variant
of the Brauer group.
41.2; Let V be an algebraic variety defined over a field k, let К D к be a
Galois extension with group G. Further, let v € И be the general point, let
k(y) (resp. K(y)) be the field of rational functions on V (resp. on V ® K).
Each element of K(v) can be made to correspond to its divisor on V ® K, an
element of the group of (Cartier) divisors Div (F ® K), The group G acts ac-
cordingly on A(v) and Div (V ® K), which permits us to define a natural ho-
momorphism of cohomology groups H2(G, A(u)*) Div(F ® A)).
Definition41.3, Br(F, A) = Ker [^(G, A(v)*)^^(G, Div(7® A))].
41.4. Because the divisor of a function measures its ‘singularities’, we can
intuitively represent the elements of Br (7, A) as those classes of algebras over
A(u) which split over A(v) and have no singularities on V. To give a precise formu-
lation of this we need the following concept:
Definition 41.5. An Azumaya algebra over an arbitrary scheme V is a lo-
cally free sheaf of Oy -algebras A on V such that the following condition is
fulfilled:
The geometric fibre A (x) = Ax ® q k(x) = AxjmxAx is a central simple
algebra over the field k(x) for every point x E V.
Informally speaking, an Azumaya algebra is a continuous system of central
simple algebras, ‘parametrized’ by the scheme И
Definition 41.6. Let A, В be two Azumaya algebras over a scheme К They
are called equivalent if there exist two locally free sheaves of Ojz-modules/Г
and F such that the (9v-algebras A ® Qy End E and В End F are iso-
morphic.
Thus the trivial’ algebras End^ E play the same role as the matrix alge-
bras over a field. The inconvenience of restricting oneself in the definition of
equivalence to matrix algebras over Oy becomes clear if one wants to retain
the triviality of the algebra A ® Qy Л0. In fact, the same arguments as those
over a field show that there exists an isomorphism of the algebra A ® Oy
with the algebra of endomorphisms of the 0 ^-module Л, which in general can
be only locally trivial over v.
Grothendieck [2] has shown that the classes of Azumaya algebras over a
scheme form a group (relative to the tensor product) which admits a coho-
mological interpretation in terms of the etale cohomology of schemes. For
our purposes only the part of this group introduced in Definition 41.3 is im-
portant. It has been constructed such that in the following investigations we
can avoid all except the more modest tools of etale cohomology theory: all
we need is concentrated in one lemma which the reader can take for granted.
J
Theorem 41.7. Let V be a smooth variety over a field k. Under the condi-
tions of 41.2 and 41.3, there exists for every element a G Br (F, K) an
Azumaya algebra A on V such that a ~ class Л (u) over k(v) (u is the general
point of F), Moreover, any two algebras with this property are equivalent
Corollary 41.8. Under the conditions of Theorem 41.7 every k-point
x G V(k) defines a specialisation homomorphism
Br(KA)-*Br(k,7Q: a^a(x).
It associates to the class of an Azumaya algebra A the class of its fibre Л(х).
We have now concluded the preparations and can state the main results of
this chapter.
Definition 41.9. Under the conditions of Theorem 41.7, let В C Br (К, K)
be a subgroup. The points xj € V(k) are called В-equivalent if for all A, В
we have а (л) = a( y).
Theorem 41.10. The В-equivalence relation is admissible for every smooth
cubic surface V over a field к and every group В C Br (К, AT) (see Definition
11.2).
Theorem 41.11. Under the same conditions we have:
(i) The CML E = V(kfB is an Abelian group of exponent a divisor of six.
(ii) If [£ : Q] < °°, the group E is finite.
Incidentally, it will be established that the quotient group Br (F, &)/Br (k, k)
is an old friend of ours: it is isomorphic to the group (Gal (k/k\ N(Vf) of
Chapter IV.
This permits an effective calculation of Б-equivalence: all of Section 45 is
devoted to an analysis of examples. We here adduce the answer to the three
cubes problem:
Example 41.12. Let к contain a primitive cubic root of unity 0, let a (V)3
and let V be given by the equation
7’^+T13 + T^+fl7’j = 0.
We set
f f TV+T2
h т + t > hr + t ’
Finally, let К - k(a*f The points x, у G V(k) are Br (V, ЛЭ-equivalent if and
only if
Arithmetical considerations then permit us to give an upper estimate of how
large the 3-groupE ~ V(k)/B is, in particular, its number of generators for
a global field k. For example, if к ~ Q(0), a G Z and d is the number of differ-
ent prime divisors of 3a, then the number of generators of E is not larger than
2d—2. All this reminds one very much of the weak Mordell—Weil theorem
and first descent for elliptic curves, and this is no accident.
41.13. The second example of Section 45 are the Chatelet surfaces. For these,
in particular, it is proved that R-equivalence coincides with Brauer equivalence.
Chatelet has proved that in his examples each R-equivalence class is param-
etrized by four independent parameters; consequently there is a clear redun-
dancy from the dimensional point of view. It turns out, however, that over
number fields two-parameter coverings obviously cannot be sufficient. Section
46 is devoted to a proof of this; it stands somewhat apart.
41.14. Finally, the last section of this chapter is devoted to violations of
the Hasse principle for cubic surfaces. In Chapter IV this principle has been
proved for some special classes of surfaces, including all minimal ones. Here
counter-examples will be indicated. They are taken from the literature
(Swinnerton-Dyer [1]; Mordell [1]; Cassels and Guy [1]). Our contribution
consists in observing that the Brauer group gives a quite general ob-
struction to the Hasse principle, and the calculations, which looked
accidental, do fit into a general theory. The idea of the construction of
this obstruction is so simple that we expound it here. We first recall some de-
finitions.
41.15. Let M be some class of algebraic varieties defined over a field k,
[&: Q] < °°. It is customary to say that the Hasse principle holds for the
classM if the following assertion is true:
Let V and Vfk^ be non-empty for all places v of the field k. Then
V(k) is non-empty.
41.16. We start with the description of the language which is convenient
to employ when introducing and classifying obstructions to the Hasse principle.
Let Ov C kv be the ring of integers (which coincides with kv if v is Archi-
medean), and let S be a finite set of places of the field k; let = {xGfcl
for all и 5}.
Each algebraic variety V has a model Vf defined over As for some S: in other
words, V' is a scheme of finite type over Spec As and its general fibre over
Spec (k) is isomorphic to V. We choose such a model V' and for a finite set
S' Э S of places of the field we set
r(*u)l 7(kv).
Here we identify F'(OV) with a subset of V(kv), The product of the ^-topo-
logies on induces a locally compact topology on The union
V(A) = U ЦЛ.,)
S’dS
with the inductive limit topology does not depend on the choice of the model
V1 and it is called the space of addles of the variety V. It is non-empty if all
FX/Q are non-empty, which we shall presuppose in the following.
The canonical inclusions k^ kv define an inclusion V(k) -> Пи and
the image of this belongs to the space of adeles, as is not difficult to check;
it is called the set of principal adeles.
In itself the condition that the vector {хи} E belong to Vfkf is ob-
tained by axiomizing the first conspicuous property of any Ar-point: the denom-
inators of its coordinates are bounded.
The majority of the classical proofs that the set K(A) is empty use the exis-
tence of some additional necessary conditions on an ad£le {xj to be principal.
The author knows of three groups of conditions, using respectively the prod-
uct formula, the non-triviality of the class group of the field к and the composi-
tion laws. The first two groups are on the whole well known from the practice
of proving that concrete diophantine problems cannot be solved, and I give
a systematic formulation only to show that a series of particular arguments
are of identical nature. The third group, however, was explicitly introduced
and subsequently studied only in the theory of Abelian varieties, where it led
to the scalar product of Cassels—Tate on the §afarevi£—Tate group. The possibi-
lity of a general formulation (and new non-trivial applications) of this group
of conditions appears to be new.
Each ‘condition’ C which we shall consider formally means the specifying
of a subset У(А)^ of the space of adeles which contains all the principal
adeles. The simultaneous fulfilment of a family of conditions Q corresponds to
the intersection ПИ(Л)С/, etc. To prove that V(k) is empty it suffices to es-
tablish that а K(4)c is empty.
The first group of conditions in their simplest formulation is as follows.
Let/GT(7, 0|r);weset
ПЛ)/={(хи) e 7(A) I n J/(*XU= »•
(The notation I I”u will be explained in Section 46.5.)
41.17. The Artin- Whaples condition: 7(k) С П {7(A)y |/G Г (K, 0j>)}.
In practice this condition is sometimes applied to rational functions/on V,
but instead of V one then in fact considers only the complement to the
support of the divisor of /
The second group of conditions does not use the Archimedean component.
Let pv be a prime divisor corresponding to a non-Archimedean place v. Using
the previous notations we set
K(4)y = {(xy) G K(4)l the divisor Пу р^&уУ) jS principal} .
41.18. The Kummer condition: 7(£) С П {K(4)y|/G Г(К 0 ^)}. This
can be summarized and generalized as follows: Let с - -> C* be an arbi-
trary quasicharacter of the group of ideles (that is, Gm(A)) of the field к which is
trivial on the principal idbles (see “Algebraic Numbers”). This means that the
product formula “ 1 holds for all x E fc*. We set
7(A )cf = {(xv) e 7(A ) I Пу c(f(xv)) = 1} .
Both of the conditions 41.17 and 41.18 are special cases of the following:
41.19. 7(k) t П cПу 7(A)y, where/G Г(V, Oy), and c runs through any
set of quasicharacters. It appears that this is a rather impractical recipe.
We now turn to the Brauer group. We recall that for every place v of the
field к there is defined the canonical embedding, the ‘local invariant’,
inv : Br к -> Q/Z ,
which possesses the following property (‘sum formula’):
2/ invv(a) = 0 for all a G Br к.
V
(For a more precise statement see Section 44.) The symbol inv^ here must be
understood to act on the image of a after the canonical base change homomor-
phism Br к -> Br kv. It must already be evident to the reader how to use this
formula for the construction of new obstructions.
Let В C Br 7 be any subgroup of the Brauer-Grothendieck group. For
every element a EB, we set
К(4)д = {(x,)G V(A)\ SJnvJzz^)) = 0}.
41.20. We have F(k) сПде5 K(4) a- An important distinction from 41.18
and 41.19 is that the variety V can be complete. It is convenient to reformu-
late this condition, so as to show firstly its connection with Brauer equivalence
and secondly its connection with the Cassels—Tate form for elliptic curves.
Definition 41.21. Two adeles (xv), (yv) G V(A) are calledB-equivalent if
(V a &B)(V v) uwv(a(xv)) = inv^ (a(ju))
(cf. Definition 41.9).
Now let E = V(A)/B be the quotient set of the space of ad Sies modulo in-
equivalence. Then every class X EE defines a character of the group В:
zy:B->Q/Z, ix(a)= 23 inv (a(x )).
Condition 41.20 is equivalent with the following.
41.22 . V(k) C Uz 0 X C V(A).
In the examples of Swinnerton-Dyer and Mordell of Section 47, we take
В = Br(F, k) for some field К and show that z*y =# 0 for all classes X.
The example of Cassels and Guy fits in by using a combination of the
Kummer condition 41.18 and 41.22.
41.23 . To conclude, we show how our construction is connected with the
Cassels—Tate form. The reader can skip this explanation without endangering
his understanding of the following.
Let Fbe a curve of genus 1 over a field k, for example a plane cubic one; let
Pg be its Jacobian. We fix on Ka structure of a principal homogeneous space
over Kg; then V defines a cohomology class h Pg (A;)), with G -
= Gal(k/k). Suppose that all the sets K(A:V) are non-empty. Then h is
everywhere locally trivial, that is, it belongs to the Safarevi^—Tate group
Ш(К0)СЯ1(С, K0O.
We now discuss the choice of a subgroup В С Br V, by means of which an
obstruction to K(£) being non-empty will be constructed. The most econom-
ical choice is obtained when E consists of one class, i.e., if for all a EB and all
places u, the local invariant invu (а(хуУ) does not depend on the choice of
In this case the unique obstruction is the fact that the character
z:B->Q/Z, i(a)= Z/ inv(a(x)),
(хи)еИЛ)
is non-zero. Therefore we choose fori? the subgroup of those elements
a G Br (K k) which are ‘everywhere locally constant’, i.e.,
(V u) a ® kv G image Br к» C Br (V 0 kv).
This is immediately connected with ILL. In fact, there exists an exact sequence
Brfc-*Br(K £) 4 HX(G, Pic ,
and an epimorphism
ф '.H^G, V0(k))^Hl(G, Pic V®k),
which is obtained from the exact sequence of G-modules
0 Pic0 V ® к = K0(I) -> Pic V ® к JXz -> 0 .
For every element a G В we have <p(a) G ф (11L (Pg)). The Cassels—Tate form
HL X JLLL Q/Z will be denoted by <, >. It emerges in our context as follows
(h is the class of V, and a GJ?):
Theorem 41,24. о ^(a)> =/(я).
It is proved by a straightforward comparing of definitions.
From our point of view, the double role of 1Л С H1 is clearly visible:
as the group which classifies the forms of Vq, and as a part of the Brauer group
of the curve V.
42. The construction of Azumaya algebras
The main aim of this section is the proof of Theorem 41.7 and some of its
consequences. We shall need:
Lemma 42.1. Under the conditions of Theorem 41.7, let A be an Azumaya
algebra on V such that its class A(v) in Br к (v) is zero. Then A is trivial.
The proof is contained in Grothendieck [2], II, Corollary l.lO.It makes
essential use of the techniques of dtale cohomology and of the smoothness
of V.
42.2. Proof of Theorem 41.7. (i) Construction of an Azumaya algebra. Let
a G Br (E K) be an element represented by the cocycle {as t}, as,t G ; we
denote by (as t) G Div (V ® K) the divisor of the function as t. Then the co-
cycle {(^ f)} GZ2(G, Div(K ® Kf) is a coboundary, i.e., there are divisors
Z> G Div (F ® A"), sGG,
such that
(ast) = s(Dt)-Dst +DS; s,t&G.
A Zariski open set U С V is called small if all the divisors Z>s ly, s G G, are
principal on U. Because it suffices to prove the theorem under the assumption
that |X: &] < we can assume that the group G is finite. Then the small
open sets form a basis for the Zariski topology on V, and every sheaf on V is
uniquely determined by its sections over these small open sets. We use this to
give an Azumaya algebra A explicitly.
Let bs E:K(y) be a local equation of Ds in the small neighbourhood U. We
set
Г(и,А) = Ф Г((/, Oy 0 K)b^ es C A(v) ,
SEG
where A (u) ~ ®seG es crossed product over k(y) corresponding to
the cocycle Д. The definition of Г07, A) does not depend on the choice of
bs, because
(bs)= (ty 'u «(b;1 b's) \u = 0 «. b;1 b's e r(t/, 0® K).
The restriction homomorphisms are defined in the obvious manner (they act
‘identically’ on the symbols es) and it is clear that U» V(U,A) is a locally
free sheaf on the small sets. Finally, the multiplication in Z(u) induces a mul-
tiplication in the sheaf because, setting e's ~ b~^es, we have
er er -a e
s t s,t st’
a = s(b ,)-1 b .b~A a ,
s,t v tJ st s st ’
and = 0 by definition, so that as t erW
The fibre of A in the general point is clearly isomorphic to A (v) as a k(v)~
algebra. It therefore only remains for us to show that the fibre Л(х) is a simple
central algebra over k(x) in any other point x G V. In fact, let x G U, where
U is a small neighbourhood; then, using the preceding notation, we have
4x)=® *;=<(*),
s6G
and the composition law is given by the cocycle {a' f(x)} GZ2(G, К ® k(x)*)
(note that Г(и, Oy ®^) = K ® Г(1Г, Oy) and that in particular
f(x) G/С k(x) for every function/G Г (U, Oy ® ^). This is not quite the
usual crossed product, because К 0% k(x) is not necessarily a field, but only
a sum of fields. However, one easily checks whether it is simple and central
over k(x) by a straightforward calculation exactly as in the case of products
over a field (Cebotarev [1], page 67).
(ii) Azumaya algebras which are equivalent in the general point are equiv-
alent. In fact, let A, В be Azumaya algebras over V such that Л(и) and B(y)
3.1 q equivalent over k(y). Then (A ® B®)(y) is a matrix algebra. Therefore,
according to Lemma 42.1, A ® ~ End E, where E is some locally free
sheaf on V. Taking the tensor product with В of this isomorphism, we find
finally
A ® (£ EndE,
which proves what we want, because the algebra В ® B° is trivial.
From the proof of the theorem we draw the following fundamental corol-
lary:
Corollary 42.3. Let f : W V be some k-morphism of smooth varieties. For
every element a G Br (К, K) we denote by A some Azumaya algebra on V such
that the class of A(y) is equal to a. Then f*(A) is an Azumaya algebra on W,
and the class off*(A) (w) (w is the general point of IF) belongs to Br (W, K).
This class only depends on a\ denoting it by f*(af we obtain a map
/* : Br(KX)->Br 0,0,
which turns Br (•, K) into a contravariant functor.
Proof. First of all,/*(4) is clearly a locally free sheaf of algebras of finite
rank. Further, for every scheme point w G W there is a canonical isomorphism
of algebras
/*(4)(w) = k(w)
so that/*(4) (w) is simple and central over k(w). Consequently/*(4) is an
Azumaya algebra on W.
Further, the algebra 4 becomes trivial at the general point of V
therefore, according to Lemma 42.1,4 ®K ™ End E, where E is some locally
free sheaf on V ® K, It follows from this that/*(4) ® К ~ End (/ ® Kf*E
on W ® K, so that the class of /*(4) in the general point w E И' belongs to
EPfG, K(w)*f An analogous argument shows that the class of/*(4) does not
depend on the choice of 4. It remains to show that it is killed by the homo-
morphism К(w)* -> Div (IE ® Kf This is a quite general fact, which in the
Grothendieck theory simply follows from the circumstance that the cohomo-
logy class which interests us is represented by some Azumaya algebra on W.
In our special situation this can be established by the following considera-
tions. Let {zzs f} K(y)*) be some cocycle in the class a. If all functions
as t are regular in the image of the general point w G W (for instance, if f is
a surjective morphism), then the rational functions f*(as t) EK(w) are defined,
and they form a cocycle {f*(as K(w)*f, and the class of this co-
cycle coincides with/*(a) in virtue of the preceding considerations. Passing
to the divisors we can write the condition for triviality in the form
L I t> A О
as above. Clearly, if f(w) is not contained in the union of the supports of the
divisors sDt (s, t G G), then
(f*(ps ,)) = -f*(Pst) +r(Ps) ,
WJ I & ij l О
so that the image of /*(zz) in the divisor cohomology is zero.
The case where/(w) is in the support of a divisor (a^t) or s(Dt) is dealt
with by passing to a cohomologous cocycle. The details are left to the reader.
42.4. Specialisation. Let/: Spec к К be a Appoint x of the variety V.
Then for an element a G Br(F, K) it is convenient to denote/* (a) G Br(&, K)
by a(x) and to call a(x) a specialisation of the class a.
Theorem 42.5. Let f: W ~^Vbea birational morphism of smooth surfaces
over k, Then the map
/*:Вг(КАГ)^Вг(Ж^)
is an isomorphism.
Proof. Every birational morphism of surfaces is a composition of monoidal
transformations with their centres at (scheme) points of V. In virtue of the
functoriality, it suffices to deal with the case of one monoidal transformation.
In this case, as is well known,
Div(W® /0=/*(Div(L® Kf)®E, Ker/*=0,
where E is the group generated by the divisors which / ® К collapses into a
point. A glance at the commutative diagram
H2 (G, K(w)*)—t H2 (G,f*(Dw(V ® KJ)) Ф H2(G, E)
H2(G, K(yf)------> H2(G, Div(K
proves what we want, because ф is an isomorphism with the first direct factor,
and the component of in H\Gf E) is equal to zero.
Corollary 42.6. Br (V, K) is a contravariant functor on the category of
smooth projective surfaces over k with the rational maps as morphisms. In
particular, the group of birational maps of a surface V into itself acts on
Kx(V,K\
Proof. Let <p : W-> У be some rational map. Its singularities can be resolved.
That is, we can construct a commutative diagram
in which g is a morphism, and/is a birational morphism. We now define the
map p* : Br (И, K) -> Br (W, K) as the composition
*
Br(r,/0 £» Br(W',K) h- BrW/C) .
The independence from the choice of/ follows from the fact that any two res-
olutions can be embedded in a commutative diagram
in which Л' and h" are birational morphisms (for this one needs to take for
(Wfrf, hft h”) a resolution of the birational map (/\)”1 о /" : W" -> PF'). After
this, some trivial diagram chasing, using Theorem 42.5 shows that p* is well
defined. Analogously, one establishes the formula о ^)* = о / in those
cases when the composition <po ф is defined. This proves the corollary.
42.7. We now remark that there is a canonical homomorphism of groups
Br (к, K) -> Br (F, K) corresponding to the structure morphism V -> Spec k.
The image of Br (к, K) consists of the classes of ‘constant’ Azumaya algebras.
This map can have a (non-trivial) kernel.
For example, the algebra of the real quaternions decomposes over the func-
tion field on the conic V: = 0. Therefore it follows from Lemma
42.1 that the homomorphism Br (R, C) -> Br (K C) is zero.
However, if there is a fc-point x on V, then the specialisation map
Br (К K) Br (к, K), a a(x)
shows that Br (к, K) is a direct summand of Br (V, K), and in particular that
no non-trivial constant algebra becomes trivial over V.
In the following important case, the group Br (V, K) is exhausted by its
constant elements.
Theorem 42.8. Let V be a smooth projective surface or curve which is bi-
rationally equivalent to P2 or pl over kt Then Br (V, K) =* Br (к, K).
Proof. One can assume that V is isomorphic to P2 or P1. Denoting by Div°
the group of principal divisors, we have in this case
Div (V ® K) = Div0 (V ® К) Ф Z .
(The projection on Z is The degree of the divisors’.) The exact sequence of
G-modules
1 K* -> Div0 (К ® K) -> 1
shows that the image of Br (к, K) in Я2(6\ A?(u)*) coincides with the kernel
of the homomorphism H2(G, K(u)*) -> Div0 (V ® Kff but this kernel
is equal to Br (V, K) by the arguments indicated above. This proves what we
want.
43. Brauer equivalence
In this section we fix some subgroup В C Br (К K). Recall Definition 41.9:
The points x, у G V(k) are called В-equivalent if a(x) ~ a(y) for all a G B. One
can obviously assume that В contains all constant classes. The following result
implies Theorems 41.10 and 41.11 (i).
Theorem 43.1. Under the conditions stated above, let Vbe a smooth cubic
surface over k. The following assertions hbld\
(i) В-equivalence on V(k) is admissible and is stronger than R-equivalence.
(ii) The CH-quasigroup E = V(k)jB is Abelian and there is a canonical em-
bedding.
T°(E) Hom (S/Br (к, K), Br (к, K)) .
Proof, (i). First of all, for all morphisms/: P1 V over k, and all elements
a G Br (К, K) and points x G P1 (k) we have a{f (x)) = /*(л) (x), and this class
does not depend on x, because the class/*(«) is constant according to
Theorem 42.8. Consequently, ^-equivalence is stronger than R-equivalence;
in particular, the classes of ^-equivalent points are dense.
Now let the points (x, y, z) and (x, y, z) be collinear on V, and x ~ x
(modT?); we shall show that z ~ z' (mod A). We consider the diagram obtained by
resolving the singularities of the birational map ty;
where / g are birational A;-morphisms.
If у is in general position with respect to x and with respect to x', then there
exist points w, w' G 1ЯЛ) such that f(w) ~ x,f(w’) ~ x, g(w) = z, g(wr) ~z-.
Then for any element a G Br (V, K) we have
а(х) = ф>Л<#)=Лл)(*')
=* U*)-1 /*(«)(*) = № /*(*)(*'),
that is, t*(a)(x) = t*(a)(*')> or a(z) =a(z’).
If у is not in general position with respect to x', and the line through x',
уt zT is not completely contained in F, then a point w' with all the necessary
properties can still be found.
Finally, if the points x', y, z are on a line in F, then they are R-equivalent,
and hence ^-equivalent. A simple geometrical analysis then shows that
there exists a point w G ^(A:) such that/(w') = g(w') = у = x (mod B) =
= zr (mod Bf and the previous arguments give the desired result.
This shows that ^-equivalence is admissible.
To prove the second part of the theorem we need two lemmas, which are
also of independent interest.
Lemma 43.1.1. For every smooth surface V with a dense set of k-points
there is an isomorphism
Br (И, K)/Br (к, K) qf Hx (G, Pic (V ® K)),
which is ftinctorial with respect to birational morphisms.
Proof. The exact sequence of G-modules
1 -+K(v)*/K* -> Div(K ® K)-> Pic(F ® K)-> 1
gives an isomorphism
Hl (G, Pic (V ® K)) 3 Ker [H2(G, K(v)* /К*) -> H2(G, Div (V ® K))].
On the other hand, the exact sequence
1 -+K*(v)-+ K(v)*/K* -> 1
shows that
H2(G, K(v)* IK*~)=H2(G,K(v)*')/H2(G, K*),
if one takes into account that the homomorphism №(G, K*) -+ H2 (G, K(v)*)
has a section ‘specialisation at a point’ and therefore has a trivial kernel. This
proves what we want.
The functoriality can be established without difficulty.
Lemma 43.1.2. Let V be a smooth cubic surface, and let x G V(k) be a good
point. Under the natural action on Br(F, /0/Вг(&, К) the homomorphism t*
is inversion.
Proof. Let W -> V be a monoidal transformation with centre x. It induces
a G-isomorphism Pic W ® К Pic V ® К ® Z and, according to Theorem
29.1, an isomorphism
Я1 (G, Pic (V ® K)) q: Я1 (G, Pic (W ® K)) .
The group Pic (W ® K) is generated by the canonical class co and the divisors
which over к consist of exceptional curves of the first kind. The automorphism
tx becomes biregular on W; we denote it by the same symbol. For every class I
of an exceptional curve of the first kind on W ® к we have
f*(D + /^~co.
Indeed for curves coming from V this is clear from the following geometrical
considerations: the class f*(/) + / is represented by the intersection of V with
the plane passing through a curve of class I and the point x, and ~ co is pre-
cisely the class of the hyperplane sections. In addition to these /, there are on
W two more exceptional curves — the inverse image of x and the inverse image
of C(x), which are interchanged by tx; their sum is obviously also equal to — co
(cf. the proof of Lemma 36.1, second step). Thus t* acts on Pic (IV ® X)/Zco
as changing signs. On the other hand, the exact sequence
0 -> Zco -> Pic (W ® X) -» Pic (W ® K)IZu 0
yields an embedding
0 -► Я1 (G, Pic (W ® KJ) -> H1 (G, Pic (W 0 /Q/Zw),
which shows that on /f1 (G, Pic (W 0 KJ) the map t* is also the changing of
signs. This proves the lemma.
43.1.3. End of the proof of Theorem 43.1. We set E = V(k}jB and define
a map
i: T°(E) -+ Hom (B/Br (k, K\ Br (k, K))
as follows: Let s G T°(B). Since В is an admissible equivalence relation, by
Theorem 13.1 (iii) there exists in the group 1?(K) of birational maps of V gener-
ated by the maps tx for all x G V(k) an element s which passes into s under the
natural homomorphism 1?(K) T(E) : tx^ tx,x G X. Further, let
h E.B/Br(k, K); we denote by h' EB some representative of this residue class
and set
i(s)(h) = s*(h')-h'.
First of all, this map is well defined. In fact, since s (and hence also s’) is a
product of an even number of maps tx,s~ acts trivially on Z?/Br(&, K) by Lemma
43.1.2, so that F*(/r') — hf G Br(fc, X). This element does not depend on the
choice of h' in the class h because F* also acts trivially on the subgroup of
constant classes. Finally, let s be another lifting of s in B(y) and let x G K(F)
be a point in which s and s are biregular. Then
s *(Л') - h’ = (s *(/f) - h')(x) = h'(s (x)) - h’(x),
in virtue of the functoriality of s *, and, similarly,
But, by definition, Л' is a function on classes of points modulo ^-equivalence,
and 5 and T induce the same permutation of these classes. Therefore z(s) (Л)
does not depend on the choice of s either.
It is obvious that z(s) (Л) is additive in h.
The map i is a homomorphism, because (using the obvious notations)
i(st){h) = t*s*(h')-h'
= F*(s *Ю ~ h’)^4h>) “ h’
= i(s)(h) + i(f)h
(taking into account that t* acts trivially on Br (к, К) С B).
Finally, the kernel of i is trivial. In fact, as above, let s be biregular in the
point x G V(k); then
У *(/f ) - У = 0« (Т*(Л') - Л')(х) = 0
&>h’(s x) = h’(x) .
If s G Ker z, then the last equality must hold for all hf GВ and for the represen-
tatives x of all equivalence classes. This means that s acts trivially on E. This
proves the theorem.
44. The finiteness theorem
Our first result is elementary.
Theorem 44.1. Let V be a smooth projective surface and let V ® к be bi-
rationally trivial. Then the group Br (К K)/Br (к, K) is finite for any Galois ex-
tension К Эк.
Moreover, zfPic(K ® К) - Pic (F ® kf then
Br (K tf)/Br (к, K) Br (K £)/Br (k, k) .
Proof. We start with the second assertion. First of all it follows easily from
the classical exact sequence of Brauer groups of fields
0 Br (k(u), K(u)) Br (fc(u)Д (u)) Br (K(y), k(v)),
that there exists a canonical embedding Br (К X)/Br (к, K) Br (У, fc)/Br (k,k).
On the cohomology groups it is induced by the inflation. which commutes
with the isomorphism of Lemma 43.1.1.
We now consider the exact sequence
. inf ч „ _
0 ->//(Gal K/k, Pic V® K) /^(Gal k/k, Pic И® Jt)
res _ _
—>//](Gal k/K, Pic V® k).
From the hypothesis with respect to the field К it follows that the group
Gal(fc/A3 acts trivially on Pic V ® к = Pic V ® K. It follows from the ratio-
nality of V that Pic V ® к is free. Therefore the last is trivial, which gives
the desired isomorphism.
As the field К one can take a finite extension of к because Pic V ® к has
finite rank, and each of its generators is ‘defined’ over a finite extension. There-
fore the group Я1 (Gal (k/k), Pic V ® k) is finite and hence this is also true
for every algebraic extension K.
Remark 44.1.1. It is not difficult to show by means of a slightly more pre-
cise argument that Br (K AT)/Br (к, K) = Br (V, k)/Bi (k, k) if V ®^K is bira-
tionally equivalent to P^.
Corollary 44.1.2. Suppose that in the group Br к there are only finitely
many elements of order dividing 6. Then the set E = V(k)!B is finite for every
smooth cubic surface V.
If in Br к there are no non-trivial elements of order dividing 6, then all
points of F(£) are B-equivalent.
The proof is obvious if one uses Theorem 43.1 and takes into account that
the relation X6 = 1 holds in T®(E\ according to Theorem 13.2. Instead of this,
one can also refer to Theorem 29.3, which shows that the exponent of the
group Br(K A?)/Br(£, K) divides 6.
This corollary is inapplicable in the most interesting case, namely, when к
is a number field. All the same, the finiteness theorem is also true in this case;
its proof, which is in itself not complicated, rests, however, on deep number
theoretical results:
Theorem 44.2. Let [& : Q] < °°. Then the set V(kfiB is finite for any
smooth cubic surface V over k.
Proof. To start, we state some well known facts on the structure of the
Brauer group of a field k.
Let p be a place of the field k, i.e., either a prime ideal in the ring of
integers of the field or an embedding or a pair of conjugate complex
embeddings k^C (Archimedean places). We denote by kp the completion of
к in the topology corresponding to p.
Further, let К Э к be some Galois extension with group G, and q some
place of the field we shall write q Ip if q induces p on k. As is well known,
к? К ; the group G which acts through the factor К on the left-
hand side, permutes q transitively and the isotropy subgroup Gq of the place
q (the decomposition group) is isomorphic to the Galois group of the exten-
sion^/^.
We now give a list of the properties of the Brauer group which we need:
(i) . If к? - C, then Brfcp - 0. If kp = R, then Br =yZ/Z; the unique
non-trivial element is the class of the quaternion algebra.
(ii) . Letp be non-Archimedean; then Br kp is canonically isomorphic to
Q/Z and Br (kp, Kq) coincides with the subgroup (1 /n ) Z/Z, where np =
- \Kq : kp\. In case the extension Kq/kp is unramified, this isomorphism is
obtained as follows. Denote by и : K* -> Z the valuation homomorphism and
construct the composed map
Br(k К К*) ~ H2( GZ) * H\G d/Z) ^-7-Z/Z.
г t t *1 Fl
p
Here a is induced by means of , and 6 is the coboundary operator, which
corresponds to the exact sequence of trivial G-modules 0 Z -> Q -> Q/Z -> 0,
and 7 is the map which to each element x H\Gq, = Hom (Gq, Q/Z)
associates the number x(F) G (1/пр) Z/Z, where J7 is the Frobenius automor-
phism, a canonical generator of the group Gq (see Serre [6]).
(iii) . There exists an exact sequence of the form
О^ВгйЛ ®p Brfc 4 Q/Z-*O.
Here the homomorphism i is the sum of the homomorphisms induced by the
embeddings к -> к? for all possible places P of the field k, and /, under the
canonical identification inv^ : Br kp Q/Z described above, is the sum of the
‘local invariants’.
This exact sequence contains in compact form the following assertions:
(a) Each element of Br к is completely determined by its local invariants.
(b) Each element of Br к has a finite number of non-zero local invariants.
(c) For each system of local invariants (..., G Q/Z (or yZ/Z,
{0}, if kp = R, C) there exists a realization as an element of Br к if and only if
ip Ф 0 for only a finite number of places p and = 0.
We now state our basic lemma:
Lemma 44.2.1. Let К D к be a finite Galois extension, and a G Br (К K)
some element. There exist a non-empty open set U С V and a finite set S of
places of к such that for all x^U (k) and p^S the local p-invariant of a (x)
is equal to zero.
44.2.2. Deduction of Theorem 44.2 from Lemma 44.2.1. It is clear from
Theorem 44.1 that it suffices to prove Theorem 44.2 for В = Br(F, K\ where
К is a finite Galois extension К D к with group G. The group Br (И K)/Br(kfK)
is finite; let a±,..., an G Вг(И, K) be representatives of its generators
modulo ‘constant’ classes. Then there exist a U and an S which fulfil the asser-
tions of Lemma 44.2.1 for all the a^ ,.. , an at the same time. From the con-
siderations in 43.1.3 it follows easily that for every element s G T§(E'} the image
of the homomorphism
ф) : Br (Vt K)IBi (к, K}^Brk
consists of the classes which have trivial invariants outside S.
In fact, suppose FGB(F)is induced by 5GT$(E); because (Vis dense, one can
choose an x G U(k) such that s is defined in x and s (x) G U(k). Then /($)
sends the class at into
so that the invariants of the image are trivial outside S.
Moreover, as remarked above, 6/(s) = 0. Consequently, Theorem 43.1 and
the result 44.2 (iii) establish an embedding
T°(E)<+ Hom (Br (К K)/Hr (к, K), |Z/Z) ,
and both groups under the Hom sign are finite. This proves Theorem 44.2.
44.2.3. Proof of Lemma 44.2.1. Let J GZ2(G, A^(u)*) be a cocycle in the
class of a. We can find divisors Ds G Div (F ® K\ s EG, such that
Denote by U the complement of the union of the supports of the divisors
s(Dr) in V ® K, s, t E G. Then for every point xEU{k) the cocycle {as t{x)}E
EZ^{G, K*) represents the class of the element a(x) E Br {к, K), and we must
investigate its local invariants.
First of all, include in 5 all the Archimedean places of к and all places in
which K/k is ramified. Then, according to 44.2 (ii), the local invariant of the
class я (x) in a place p ES is represented by the class of the cocycle
(^(aIfWF22(G Z),
ч 4
where q is any extension q Ip of the place p on K.
We must now use the central result of A. Weil’s theory of distributions
which we shall describe in a moment. The result cited below is obtained imme-
diately from Theorem 10 of Weil [1], with the obvious changes in notation.
Proposition 44.2.4. Let Fbea group of divisors on V ® К whose supports9-
are contained in the complement of a non-empty open set U С V ® K. Then
there exists a family of homomorphisms
indexed by the points xEU(k) and the non-Archimedean places q of the field
K, with the following properties'.
(i) V (/) EF is a principal divisor, then for all except a finite number of
places q of the field {the exceptions depend on f) and for all points x E U(k)
we have
4>л 4
(ii) For alls EG ~ Gal K/k and DEF,
We apply this result to the divisors (as f) which occur in the cocycle
{zz5 f}EZ2 (G, AT(v)*). Because there are a finite number of them, we can en-
large the set 5 of places q (while it remains finite) such that the assertion
44.2.4 (i) is true for all (as t). As a result we obtain that for q S and xEU(kf
= A x(^z) - \ x(Pst)+ А •
Restricting ourselves to the case s, t G Gq and using 44.2.4 (ii), we obtain
A x = A* x(sDJ=A xW •
Therefore for xGUfk) and q $ S the local invariant of the class a (x) is repre-
sented by the cocycle
{A xW - A /АP+ A x(°s)^ez2(A>z) -
which clearly is the coboundary of the cochain
{Ax(A))GC1<A’z)-
This concludes the proof of the lemma and of the finiteness theorem.
Corollary 44.2.5. Under the conditions of Theorem 44.2, В-equivalence is
trivial on V(kp) for almost all places p of the field k.
I do not know whether the analogous result holds for universal equivalence
or R-equivalence.
45. Calculations for Brauer equivalence. Examples
In this section, summarizing the preceding discussion, we shall describe an
effective algorithm to determine whether two given points x, у G V(k) are
В-equivalent. Afterwards we shall apply this algorithm to concrete examples.
We start with calculating the Brauer group. We keep the notation of the pre-
vious section.
Proposition 45.1. Let F be a G-invariant group of divisors on V ®K, the
classes of which generate Pic (V ® K). Then
Br(V,K)/Br(k,K) ~ Ker[H2(G,L/K*) ^H2(G,F)],
where L CK(v) is the group of rational functions on V® К of which the
divisors belong to F.
Proof. The commutative diagram of G-modules
1 —------> L/K*--------->F--------->Pic V ® К------>1
1 ----------------->Div V ® К------>Pic V ® К----->1
yields the commutative diagram
0 -+Hl(G, Ис V ® K)---->H2(G, L/K*)----->H2(G, F)
O^hHg, Pic K® K)—+H2(G,K(v)*/К*)-*H2(G,DivV®K)
An appeal to Lemma 43.1.1 proves what we want.
4 5.2, We can now proceed with an informal description of the algorithm
for the calculation of Brauer equivalence.
(i) . Choose a finite Galois extension K/k and a finitely generated group
of divisors F, generating Pic V ® K, with a finite number of generators.
The typical situation is: For F, take the group generated by all the 27 lines
on V ® k and for G a finite normal extension of k over which all these lines
are defined. Sometimes one can do with only part of these lines and a corre-
spondingly smaller field K.
(ii) . Calculate a finite number of cocycles
{c№}&Z2(G,LIK*)
ij, *
of which the classes generate
Ker [H2(G,L/K*)~>H2(G,F')].
Because G is finite, and F and L/K* are finitely generated groups, this step
can be carried out effectively. In practice it is usually convenient first to cal-
culate the cocycles which represent a system of generators of the group
H1 (G, Pic V ® K) ~ Hl(G, F/Fo) ,
where FQ C F is the subgroup of principal divisors, and to map them after-
wards into L/K*) by means of the coboundary operator as in Proposi-
tion 45.1.
(iii) . Construct functions{as t}EL such that (mod AT*) (the
have been described in the previous step). It is not necessary for this to care
about whether the Д generate cocycles of C2(G, £*).
(iv) . The final result: Suppose that the points x, у G F(k) do not belong to
a support of the divisors of the group F. Then x ~ у (mod B) if and only if all
of the following cocycles represent zero:
— ez2(G,K*), i=l,...,r.
4' M
A
In turn, to check whether a given algebraic number theoretical cocycle is triv-
ial reduces to a finite number of local calculations.
(v) . An important particular case: G is a cyclic group. The calculations are
simplified in this case thanks to the existence of an isomorphism (which de-
pends on the choice of a generator of G)
H2(G. K(y)*/K*) ^H°(G, K(y)*/K*) ~ k(vf/k*N(K(vy) ,
where N denotes the norm of K(v) into k(y). In this case each cocycle {a can
be replaced by a single element b® E k(v)*/k*N(K(v)*), and for that purpose
one can take a representative of this element,
Two points x, у G V(k) which do not belong to a support of the divisors (tW)
will be equivalent if and only if
b®(x)
£V*)>
/= 1,.
Example 45.3. Let char/: Ф 3, a E k*^ (k*)\ and = 1, where 0 Ek is
a primitive root; let К ~ k(aTf G = Gal K/k. We consider the surface
И: Tq+t3 + tI + аТ% = 0,
and rational functions on it,
f Л+вТ1 J0+^
J\ ТУТ ' ^2 T + T ’
70 yo
(these notations will be kept up to and including 45.7.5).
We shall prove the following result:
Proposition 45.4. The points x, у G K(&) at which T§ + Т^ Ф 0 and
To + 72 #= 0 are В-equivalent if and only if
Proof. According to the recipe given in the beginning of this section, denote
by F G Div V ® К the group of divisors generated by the lines on V ® K, and
by Fq G F the subgroup of principal divisors. A choice of a generator in
G = Z3 yields the commutative diagram
tf2(G,F0) -> H2(G,F)
H°(G,F)
F^/NF0 —-> Fg/NF
where N = 2^$$. This diagram permits us to identify the group
Br(U, K)lBi(k, K) ,
the kernel of the upper arrow, with the group
F^NF/NFQ,
the kernel of the lower arrow.
We now need to analyse in thorough detail the groups Fq, F and the action
of G on them; we start therefore by introducing some convenient notations.
Let к =0,1 or 2, let (z, /) = (0, 1,2) ^(k), i <Л and let m, n run independently
through a full system of equivalence classes mod 3. The 27 lines on V ® К have
the form
^Ti-vBmTj =0,
LK(w,n):
It+0"^t;=o
(a* EK is fixed once and for all). Further we put
M (m) =N(L (m, и)) ; T. + 0m T. = 0 .
The nine divisorsturn out to be free generators of the group NF. Be-
cause they are plane sections, they are pairwise equivalent (linearly), so that
the group Fq П NF is generated by the pairwise differences of these divisors.
Further, the divisors
Ц)=ТИ2(1)-Л/2(0),
(Г2)=м1(0)-л/2(0)
belong to Fq Ci NF, and for the proof of Proposition 45.4 one needs to verify
that their classes constitute an (independent) system of generators for the
group Fo О ^F/WQ.
Firstly, by means of considerations on invariants, we make sure that there
are precisely two generators. The following result can be derived from Table 1
in Section 31 or calculated by means of Propositions 31.3 and 31.5. For com-
pleteness’ sake we give another variant of the proof, taken from Safarevic’s lec-
tures [1].
Lemma 45.4.1. Я1 (G, Pic (V ® K}) = Z3 X Z3.
Proof. We recall the definition and fundamental properties of the ‘Herbrand
quotient’. Let G be a finite cyclic group, A a G-module such that the orders
h*(A) of the groups H*(G, A) (Tate cohomology) are finite.
The known periodicity ZF+2 = Я7 naturally suggests the introduction of
the following variant of the Euler characteristic
h(A) = h°(A)/hl(A).
h(A) is called the Herbrand quotient of the G-module A, and it has the fol-
lowing properties:
(i) . Let 0->Л~>7?~>С~>0Ьеап exact sequence of G-modules and sup-
pose that two of the three numbers h(A), h(B), h(C) are defined. Then the
third is also defined and
h(B) = h(A) • h(C) .
(ii) . Let Л J, A% be G-modules which are free and of finite rank over Z. If
the G-modules R an^ ®Z^2 are isomorphic, then h(A 0 = #(Л2)
(see the book ‘Algebraic Numbers”, Ch. IV, § 8).
We apply these properties to the calculation ofZ/^G, Pic V ® K\ Because
V is minimal, (Pic (K ® K))G = Zgjbut gj у is the class of a hyperplane .
section; in particular, it contains the norm divisors MK (m). Therefore
H°(G, Pic V ® K) = (Pic V ® K)G/N (Pic V ® К) = 0 ,
and
Л1 (Pic V ® K) = h (Pic V ® a:)-1 .
Because the exponent of H1 divides 3 (the exponent of G), it suffices to prove
that h1 = 9, i.e. h = |.
The space R ® z Pic V ® К is seven-dimensional; the group G acts on it, and
the identity representation turns up with multiplicity one in this representation.
Therefore, setting I - Z [G] /Z, we have
R ®z Pic V ® К ss R ф (R ®z I)3
(as G-modules), because R 0Z Z is the unique irreducible representation of G
over R different from the identity. Using properties (i), (ii) of the Herb rand
quotient, we therefore find
A(/) = A(Z[G])/ft(Z) = A(Zr1 ,
й(Ис V ® К ) = A(Z) h{I)3 = A(Z)-2 .
But H°(G, Z) =s Z3 and H1 (G, 2) = 0, which concludes the proof.
We now return to the analysis of the configuration of the lines.
Lemma 45.4.2. Let Vbe a smooth cubic surface over a field К over which
all its lines are defined. Then the group of principal divisors, consisting of lines,
is generated by the pairwise differences of those plane sections which decompose
into three lines.
Proof. We choose six non-intersecting lines on V and collapse them by means
of a morphism/: V P2. We introduce the following notation for the lines on V
(see Theorem 26.2): p^ ,...,p^ are those lines such that the/(pz) ~p^ are points
on P2;I(i,j = 1,..., 6; z’ =/=/) are those lines such that the f(ljj) - Iare lines
on P2 passing through the points pir pp q^i = 1 ,.. . , 6) are those lines for
which the f{q^) -qi are conics which do not pass through the point p^
With this notation, after some practice, we can easily see which lines inter-
sect. For instance, /12 does not intersect Z13 because and Г13 intersect on
P2 at the point which gets blown up and ‘throws apart’ these lines. Simi-
larly, /13 does not intersect but does intersect and <?3 (only one of the
intersection points of Г^3 and q^ on the plane is thrown apart)..The plane sec-
tions on V are exactly the triples of pairwise non-intersecting lines, .e.g., <?z + pj +
+ Iii (z Ф/) and 7zy + I + lrs, where (z jp q r s) is a permutation of (1 2 3 4 5 6).
. Further, Fq represents the inverse image under/of the group of principal
divisors on P2 with components and ~qK. This group is clearly generated by
divisors of the form
hj ~ ~ ~ ^23 “ ^45 *
Taking into account that/*(7^.) = T + pz + /*($z) “ qt + we find for
Fq the system of generators
lij-lrS+Pi+Pj-Pr-PS’
4i ~ dj ~ Pi + Pj = (Я, + Pj + Pj) ~ (df + Pi + ty ,
^1 " Z23 “ Z45 +^6 ” +^6 + Z16^ “ ^16 + Z23 + W *
The divisors of the last two types have already been represented in the form of
a difference of plane sections. As regards the first type we remark that firstly,
for i = r,
~ lis *Pf-Ps = + ks + PS) ,
and secondly, for (zy) A (rs) = 0,
Г..-Z = (Г.._д) + (Г - Г ),
l] rs V IJ ISJ v is rsJ ’
which reduces the matter to the previous case and concludes the proof.
From this lemma we deduce:
Corollary 45.4.3. Under the conditions of 45.3,45.4, the group NFq is
generated by the pairwise differences of divisors of the form
3M(m), к, m = 0,1,2 ,
К
2
S M (m), к =0,1,2.
/С
m~0
2
m = 0,1,2.
К
Proof. We check that the divisors described are norms of all the plane sec-
tions contained in F.
First of all, a straightforward calculation shows that each line L^{mf ri) in-
tersects precisely nine lines -
LK(m,n'), LK(m',n), L.(m2,n2),
where ri Фп,т Ф m, and the indexes nlf m2t n2 satisfy the following
conditions:
= m +n (mod 3) if к
m2-n2
m2 + n2
= m +n
= m. - n
= n-m
(mod 3)
(mod 3)
(mod 3)
if к > j,
iii<K ,
if i > к .
These nine lines in turn split up into five connected pairs, which together with
Ьк(т, ri) constitute all triples of coplanar lines containingLK(m, ri).
The first of the divisors described in Corollary 45.4.3 is the norm of the
triple S^=0ZK(m, и); the second the norm of the triple n) and,
finally, the last three are the norms of the remaining three triples, which the
reader can check himself with little trouble.
Corollary 45.4.4. We define the homomorphism
X:Fon7VF^Z3XZ3,
by setting
x(Sa M(m)) = (S K-a (mod 3); S ma (mod 3)).
Л7 f t Л Л- It L Л- III
Then Ker x “ NFq, and because
X((/i)) = (0,l), x((/2)) = (-l,O),
the classes (Д ) (mod AF0) and (/2) (mod 7VF0) generate Fq П NF/NFq .
Proof. It is immediately clear from the formulae of Corollary 45.4.3 that
NFq C Ker X- On the other hand, (Д) and (/2) generate in Fq Л NF/Kei x a
subgroup of order 9, therefore Fo П TVF/Ker x, as a factor group of
Fq DNF/NFq — Z3 X Z3, must coincide with it. This concludes the proof of
Proposition 45.4.
We now apply it to number fields.
The local case. Let : Qp] < °°, ? : k* -> Z a valuation (equal to one on a
generator of the maximal ideal), and let A C k be the ring of integers, m its maxi-
mal ideal and U = A ni the group of units.
Proposition 45.5. If a is an integer and За E U, then for all points x G V(k)
in which + T? 0, T? + Tf Ф 0, we have
XJ X JI At
f2{x)^NK/k(K*).
In particular, all points of K(/c) are Br V-equivalent.
Proof. Letx = (Zo, Zp f2, *з)> w^ere the GA are integral and relatively prime.
From the fact that 3a G £7 it follows that the extension К = к(а*) is unramified
Therefore
NK/k^ = G ГI u(f) = 0 (mod 3)}.
We now examine the functions Д and /2 separately, distinguishing in turn the
possible cases.
Study of f^x).
(i) v(t0 + fj) = v(t0 + 0^). Then и(/\(хУ) = 0 =>Д(х)&NKjk(K*).
(ii) r- v(tq + Zj) ¥= l>(/q +0/j) = s. Then u(/q) - w(7j), and therefore
uOq + ~ min (r, s). Because
2
П (r0 + f)‘t.) +N(t2 + fih3) = 0 ,
!=0
we obtain r + s + min (r, s) 0 (mod 3), so that
(*)) = r - s = 0 (mod 3) (x) G NK/k (K*)
Study of f2(x).
(i) v (r3) = 0. Then v(tj + fj) = 0 for all i Ф j, i, j G (0, 1,2) (from which
follows u(/2(x)) = 0 =>/2(x) G7V^(/C*)). In fact, in the opposite case we
would have for к Ф i, к Ф /, к G (0, 1, 2),
a = (—t* Z^l)3 (mod m )
( in G A is the maximal ideal). Because 3a G Ut it would follow from this that
a G (A;*)3, in contradiction with the assumptions.
(ii) v(?3) > 0, v(t$tyt2) > 0. Because t$, t2, £3 are relatively prime,
for precisely one index i G (0, 1, 2) we have v(/z-) > 0. If v(fj) > 0, then
и(Г1 + f2) = u(f0 +1{) ~ 0 => n(/2(x)) = 0. If u(/-0) > 0, then v(t1) = v(t2) = 0,
u(t0 + Z^) = 0 and one must separately consider the case uCfj + z2) > ® only.
But then
1>(L + 0Z9)=u(Z. + 0“Z?)-O
Lx* L x
(we here use the invertibility of 1 — 0, a divisor of 3, in A), so that
+ Z2) ~ + t?) “ u(f2 + at3^ 0 (mod 3)
and
v(/2(x)) = 0 (mod 3) .
The possibility u(z2)> 0 is dealt with in entirely the same way.
(iii) v (/3) > 0, v(?q^ Z2) = 0. Then v(tj + Z? ) = 0 for all i Ф/, it j G (0,1,2)
(otherwise v(tK) > 0), so that u(/2(x)) = 0.
This concludes the proof.
A particular case, к = Q2(0), a = 0. All the conditions of Proposition 45.5
are fulfilled, so that Br К-equivalence is trivial. On the other hand, it was
demonstrated in 16.3 that reduction mod 2 defines a non-trivial admissible
equivalence relation on the surface И
Consequently, in this example Brauer equivalence is strictly coarser than
universal equivalence.
The global case. Now let [ft : Q] < °°, let a €E к be an integral algebraic
number, let p run through the non-Archimedean places (i.e. the prime ideals)
of the field A, and let Ea = 7(k)/Br V.
Proposition 45.6. Under the conditions described, the construction of
Theorem 43.1 yields an embedding
i: Hom (Z3 X Z3, ®'p| 3a |Z/Z) ,
where the prime on the direct summation sign means that we consider the sub-
group of those vectors whose components add up to zero.
In particular, let rk Ea denote the number of generators of the Abelian
group T^iEf) (or of Ea with composition law U о (X о У), and let d be the
number of different prime ideals dividing 3a. Then гк-£д < 2 J - 2.
Pfroof. Using the notation of 45.3 take as h\, G Br (V, K) the classes
of the algebras over V corresponding to the functions Д, /2. Proposition
45.5 it was proved that if рКЗа and a ^(k*)3, then the local p-invariants of
the classes of Л|(х), k2(x) G Br(kp) remain invariant under changes of x in
V(kp). This is also true for a G (k* )3 because the surface Vk ® k? is bira-
tionally trivial in that case. The Archimedean places do not contribute to
Br к because 0 Ek and hence the field к is purely imaginary.
An explicit treatment of the construction of the embedding i given in 43.1.3
now immediately shows what we want.
45.7. We now state some unsolved questions.
45.7.1. Let cx, c2 G k* and [k : Q] < °°. Do there exist points x G K(k)
for which ffayECjN%(K*), i = 1,2? It can be assumed that the necessary
local conditions are fulfilled.
45.7.2. To what extent can Br И-equivalence be different from R-equivalence
and U-equivalence in this example?
45.7.3. Is there a bound for rk (Ea) as a runs through all possible numbers
of the global field k? This, clearly, is analogous to a well-known unsolved prob-
lem in the theory of elliptic curves. For a discussion for the curve Tq + T3 +
+ aT2 = 0, see Cassels [2].
45.7.4. Finally, it would be interesting to examine the case when в к
(for example, к - Q). Adjoining 0 gives a quadratic extension k(0), and on
K(k(0)), Br F-equivalence is defined. It induces some admissible equivalence
relation B' on U(k); moreover, it is probably possible to define a norm map
^У(т/вту^у(к)1в(
(see the construction in Section 15 concerning R-equivalence.)
45.7.5. Do there exist connections between rk(£0) and the zeta function
of the surface
We now pass on to the analysis of another example.
Example 45.8. We consider the surface studied by Chatelet [1], [2]. Let
char (к) Ф 2, a € k* \ (A:*)2, let ay, a^, a^Ekbe three different elements,
and let К = k(\/af The surface V is given by the equation
V- TQ(Tl-aTl)=h(T3-aiT(iY
i=\
On V we consider the rational functions
, T3 - “lT0
l~T3-a3T.’
- - Гз
This notation will be retained until the end of this section; let В = Br(K, Kf
Proposition 45.9. Two points x, у E V(k) in which - afT§ =# 0, Zq =# 0,
/= 1, 2,3, are В-equivalent if and only if
^х)!^у)^К/к(К^, i=l,2.
Proof. Before starting the calculations, we point out one difficulty:
the surface V has two singular points (which are conjugate over k) on
the hypersurface at infinity Tq - 0: Ty ± T^\fa - 0. Therefore, under Brauer
equivalence on V we must understand the equivalence relation induced by B-
equivalence on some non-singular model of V: this does not depend on the
choice of the resolution of singularities according to Theorem 42.5.
45.9.1. The construction of a non-singular model. In the direct product
pl X P1 X P1 with threefold homogeneous coordinates (Xq, Xy, Уо, Ур Zo,
Zy) we define a surface Wo by the equation over k:
^0 : Zl<Z0 K1 = WZ0 - a2Zl)(Z0 - a3Zl) •
An easy local computation shows that PP0 is smooth for this can
be assumed without loss of generality because a non-singular linear transfor-
mation of (Tq, T3) permits us to obtain this.
There exists a birational morphism over K,
^wv®kK-*v®kK>
which on the generators of the field of rational functions K(v) is given by the
formulae
(T \ z
211 = 12
T J Z
V 1
The surface has a form W over к and a birational map : W -> V over
к such that <p0 = <p ® K. To describe W, we denote by t: Hq JVq the auto-
morphism of order 2 induced by the permutation of the first two components
ofP^XP1/?1:
t*(X^Y., t*(Yj) = X., t4Zf) = Zit z-1,2.
Further, let Z2 act on the A>scheme 1VO ®k К as {id, t] on the first factor and
as G = Gal (K/к) on the second factor, and let Z2 act on V ®^ К as the identity
on the first factor and as G on the second. The map is then compatible
with these actions, and that permits us to descend it to a map
: W = (Wo ®k K)/Z2 -+ (V K)/Z2 - K.
The surface If, being a form of JVq, is smooth. It is also the model of V on
which we shall carry out the calculations. We identify W ®k К and Hq ®k K,
and also the divisor groups of these surfaces.
45.9.2. The geometry of W. To analyze the structure of the surface W and to
choose a group of divisors F C Div (If ®k K) such that their classes generate
Pic (If ®k K), we consider two projections of W С. P1 X P1 X P1: and p13.
(i) The projection p3 : W-+ P1 on the axis (Zo, Z^f The fibres of this
morphism over the points (1,0), , 1), (a2, 1), («3,1) consist of two lines
which intersect each other transversely. Over any other point the geometric
fibre is isomorphic to P1.
This morphism has two sections: the curves/)1: = Kj = 0, and
D^: Xy = Yq - 0. The configuration consisting of the two sections and the
components of the special fibres is indicated in Fig. VI. 1. The fibres have been
numbered in the order in which they were enumerated in the previous para-
graph. The pairwise intersection numbers of these curves, equal to 0 or 1, are
easily discovered from Fig. VI. 1. The self-intersection numbers turn out to
be as follows: (Dj, £>/) = -1, (Z>4 = —2. In particular, the P/ are excep-
tional curves. The first equality follows from the fact that the fibres Z>? + Z>?
are pairwise equivalent to one another and do not intersect; consequently,
(£>}, + D2) = (P], Db + 1 = (pl D? + О?) = 0 , j*i.
Ill L J f
The second equality is deduced from the equivalence of the divisors
(Хо=О)=л! ,
(Xj = 0)=D2 + D2 + D2 ,
from which
(Z>1,P1) + 2 = (P1, D1 +DlQ +D\)=(Dl,D2 +D2 +D2) = 0.
Finally, the generating element of the group G= Gal K/k interchanges the places
of D1 and D2 and the components of the fibres, sending each fibre into itself.
Fig. VI. 1.
To calculate (G, Pic W K), it is necessary to know that Di and D!
generate the whole Picard group. To prove this we consider one other projection:
(ii) The projection p^ : W -* P1 X P1. This morphism has the following
properties, the verification of which we leave to the reader as an easy exercise:
(а) р^з is a birational morphism which collapses the exceptional curves
Dl into Points (0, 1, 1,0), (0, 1,^, 1)(1, 0, 1), (1»0, a3,l),
respectively.
(b) The monoidal transformation of P1 X P1 with as centre the union of
the four points listed above is isomorphic to p13.
We now observe that the curves (7 =р13(Рг) are fibres with respect to the
two projections of P1 X P1, so that their classes generate Pic ((P1 X P1) ® k).
It follows from this that Pic W ® к is generated by the classes of
p*3(C1)=JD1+^+^,
p*3(C2) = £>2 +D\+D^,
In particular, let F be the group of divisors on W ® К = Wq & К generated
by all the D\ Then the canonical map F Pic (W ® K} is suqective.
Moreover, property (a) shows that the surfaces Wt PV0, V become birationally
trivial over K.
45.9.3. The calculation о/Вг(И4 X')/Br(^, K). Let Fq C F be the subgroup
of principal divisors. Because G = Z2 is cyclic, we obtain as in Proposition 45.4,
Br(H< /C)/Br(^,/C)^F0 C\NF/NFq , 7V=l+s, G = {1, s}.
Let D G Fbe any divisor; because Fgenerates the Picard group, and because
numerical equivalence coincides with linear equivalence on rational surfaces,
we have
Z)€R ф>(Д -(£>,£>/) = 0 forall /=1,2, / = 0,1,2,3.
V *
Therefore, using the table of intersection numbers, we obtain without diffi-
culty that
hdl=2d
z-0 J
On the other hand, ND? - ancj Д79/ = £)1 + Z>^ Consequently,
f 3 3
NFn = 2 S d.(D] + D?) + 2d Zz D?
(j I —• zv z z 7 z
\ z-0 z-0 ;
3
Tjei = 0
z=0
We define the map
X:FO О NF-+Z2 X Z2
by the formula
/ 3 \
x(Z/e-(Z\- + £>?)!- e3 (mod2), e2 -e3 (mod2)) .
\/=0 /
One can easily convince oneself that x induces an isomorphism
Fo nNF/NFQ^Z2 X Z2 .
In particular, this group is generated by the divisors of the functions
(Z a z \
7 t n=(z?l +jD2) _ (£)1 +732) t
(Z u Z \
/-ЛГ(Д2+Д1)-(Д3+Д1)'
which also concludes the proof of Proposition 45.9.
In contrast with Example 45.3, we can make the following essential supple-
ment:
Proposition 45.10. On a Chatelet surface К Brauer equivalence coincides
with ^equivalence. More precisely, for every В-equivalence class XC V(k)
there exist a four-dimensional special manifold W and a k~morphism
such that is a thick subset of X {that is, on a dense open subset
UCVwe have U П X = U П W)))-
Informally speaking, the points of one B- (or R-) equivalence class on V are
parametrized by four independent parameters.
Proof. It is sufficient to establish this assertion for one equivalence class;
we take the class Xq for which ffa)ENK^{K*), i~ 1,2, for all x G Xq . To
construct W, we use the construction of 15.1.3. That is, we set V~R^^-{V®K).
Since the surface V ® К is birationally equivalent to P^, the four-dimensional
A>manifold V is birationally equivalent to P^. Further, we identify V(k) with
K(X). There exists a birational map over к, V X V, such that for every point
x G V{k) - V{K) of which the conjugate (over k) x is in general position with
respect to x, we have
Ф(х)=^хох gK(fc)cK(X)
(see the construction of the norm map in 15.1.3).
Let W С V be the dense special subset on which ф is defined, and let
tp : W -+ V be the restriction of ф to W, Because ^-equivalence is stronger than
R-equivalence, Proposition 45.10 immediately follows from the lemma:
Lemma 45.10.1. <p(W(£)) is a thick subset of the class Xq.
Proof. Following Chatelet, we shall establish this by a direct calculation.
Let x E Xq C V(k) and suppose that in the point x the sections Tq, T^— afpQ
do not become zero. We shall prove that then there exists a pointy in F(X) =
= V {k) such that у о у = x. It is clear that the lemma follows from this because
</?(IP(^)) is completely contained in one В-equivalence class.
We set f = T^/Tq, i~ 1,2,3. The argument consists of three steps:
(i) x G Xg 4=> t^(x) — at CNKfk{K*) for all i = 1,2,3. In fact, because
f{x) ENg/kfK*), by definition there exists an element c G k* such that
r3(x) — a^ G cNKfe(K*y, moreover,
3
П (t3(x) -az) = t3(x)2 - at2(x)2 EN^K*) ,
so that c3 ENKfe(K*)^c ENK/k(K*). The opposite implication is obvious.
(ii) There exists a pointyr G К (ft) such that;/ andy' are in general position
and such that the -coordinates ofy' о у' and x are the same:
^3(x)^3C/oj') .
We shall indicate the coordinates of such a point y’ explicitly. We set
tJx)-a. = w?- avj=N(wf) , w. = u. + y/av. EK, z = 1,2, 3 .
•z ILL L L L I
Further, let
X = ) jj. = Wj - w2 .
We construct a pointy' with coordinates ^(y'), ^(./)Дз(./) which we find
from the equations
^(Л+мЦО'') =Xw2^2’
^У'У-'/а^у") = A<W3^3’
%O') =a1+\jU-
It is easy to check that у' G V(k). It is more difficult to verify that
t^(y! ° у') = f3(x). Chatelet arrives at this after having calculated all the
formulas for the coordinates of the pointy' о у ' explicitly. I cannot suggest
anything better and therefore omit these rather cumbersome direct calcula-
tions.
(iii) Conclusion of the proof We consider now the affine curve Г defined
over ky by the equation - ar2 - 1. This is a group with unit element (1, 0);
it is the multiplicative group under the classical composition law of a Pell form.
It acts k-linearly on the surface V, preserving the projection of V onto the
f3-axis. Under this action, a pair of points x G V(k\ z G Г(£) corresponds to
a point zx G K(&) of which the coordinates can be found from the formulae
r3(zx) = r3(x),
ty (zx) +\/at2(zx) = (fj (x) + y/at2(x)) (r* (z) + \/ar2(z)) .
The action of Г(&) on the set of A>points of each irreducible fibre of the f3-
projection is transitive. In fact, the ratio of two numbers of К with the same
norm in к has the norm of a unit (we restrict our considerations to an affine
part of F). Therefore there exists a point z E Г(&) such that z(yf о у') - x.
On the other hand, from the ^-linearity of the action of z it is obvious that
Z(y' °y')=(zy')o (zy').
Consequently, the point zy ~y fulfils the conditions of the lemma. This
concludes the proof of Proposition 45.10.
Corollary 45.10.2. The CML E ~ V(k)IR is an Abelian group of exponent 2.
It is finite if к is an algebraic number field.
We shall now study the number field case a bit more in detail.
The local case. Let [£ : Qp] < 00 and v : k* -> Z a valuation. Let А С к be
the ring of integers, and U = A* the group of units.
Proposition 45.11. If a, ai are integers and 2a — aj) E U, then for
all points xEVfk) for which T§ Ф 0 and T^ - afT§ Ф 0, we have
/-1,2.
In particular, all these points are B- and ^-equivalent.
Proof. Let x - (r0, t±, t2, /3), where the f EA are relatively prime. The ex-
tension К = k(y/a) is unramified because 2a EU, so that
e I s 0 (m°d 2)} .
The functions/* and f2 are transformed into one another if one renumbers
the лг-; therefore it suffices to deal with/Дх). We consider two cases separately:
(i) v(t3~axtQ) =v(r3-a3Zo).Thenv(/1(x)) = O=>/’1(x)e7V^/fc(A’*).
(ii) r = v(r3 - ay r0) =# v(r3 - a3t0) = s. In this case
v(tQ) = n(r3) = u(f3 - a2t0) = min (r, s) .
Take into account that - at %)= (t$ - - а^ц) and compare the valua-
tion of both parts. We obtain
min (r, s) + 2q = r + s + min (r, s) .
Consequently, r = s (mod 2), so that v(f(x)) 0 (mod 2).
This proves the proposition.
We remark that if к = R, we have E = {1} or Z2, depending on whether
K(R) is connected or not, according to Corollary 16.1.3.
The global case. Now let [& : Q] < 00, let a, a^ G к be integer algebraic
numbers, let p run through all possible (Archimedean and non-Archimedean)
places of the field k, and let E = V(k)/B = V(k)/R.
Proposition 45.12. Under the conditions described, the construction of
Theorem 43.1 yields an embedding
i: T°(£)^Hom(Z2 X Z2, ©^Z/Z) ,
where S consists of the real Archimedean places for which V{kp) is not con-
nected, i.e., p(a) > 0, p(af) G R, and the primes dividing 2a TI (af - Oj). In par-
ticular, let xkE be the number of generators ofT®(E), let d be the number
of different prime divisors of 2 я II (a7- - ay), and let n\be the number of real
places p : к R/or which p(a)>§. Then
rkE<2(d + n\)-2 .
Proof. One repeats the arguments of 45.6 with the obvious modifications.
We note that in the construction of Example 16.2 we made use of pre-
cisely the ‘Archimedean part’ of this result as we did not yet have available
the general theory.
Concerning the Chatelet surfaces one can also pose questions analogous to
those of 45.7.1,45.7.3 and 45.7.5. In particular, let cx, c^ &k* be ele-
ments such that their product ^iC2c3 There exists a point
x G V(k) in the class for which f(x) G (К*), i = 1, 2, if and only if the
system of equations (see 45.10.1 (i))
t3-ai = ci (u? - a-v?) , i = 1, 2,3 ,
is solvable in k. Eliminating from this, we obtain the four-dimensional in-
tersection of two quadrics in P6:
Q'
cl^wl ~ ~ ~ a2u2^ = ^a2 >
el(ul “ ез(из ~азиз) = (аз ~a\) •
Conversely, starting with ^-points on Q, one can construct points on Kin the
corresponding R-equivalence class (this is clearly not a 1-1 connection, in
particular with respect to the choice of the sign of ty t2).
If Q(kp) is non-empty for all places of the field k, does i; follow that Q(k)
is non-empty? This is a typical problem arising in connection with ‘first de-
scent’.
46. A negative result
In the last example of the previous section, for Chatelet surfaces V qnqt
number fields k, we obtained a rather complete qualitative description of the
structure of the set of rational points of V(k). Let us recall it once more in
outline:
(i) V(k) decomposes into a disjoint finite union of classes with
respect to R- (or Br K-) equivalence.
(ii) These classes form an Abelian group E of exponent 2 with respect to
the composition law U о (X о У).
(iii) Each class (more precisely, a thick subset of it) is covered by a
morphism of a four-dimensional special variety.
Apparently it was precisely the unusual character of this description in
conjunction with the many calculations in Chatelet’s work which caused the
fact that his construction did not attract attention for a long time. Chatelet
himself gave the decomposition E in terms of the functions fy /2 by analogy
with the elementary proof of the weak Mordell—Weil theorem, and he verified
all properties of the composition law and of the classes by means of explicit
formulae.
From the point of view of our general theory, all characteristic features of
Chatelet’s construction get an invariant explanation. In particular, E is a group
of exponent two, because V becomes birationally trivial over a quadratic exten-
sion of the field A; (Proposition 15.2). Moreover, this group is a birational invariant
of V over к and it does not depend on the choice of the model and formulae.
It can have arbitrarily large order (Example 16.2) but is finite over number
fields: this is always true for Br K-equivalence.
The only embarrassment may be raised by (iii): Why does one need to use
four parameters to describe the rational points of a two-dimensional manifold?
The main aim of this section is to show that over number fields two param-
eters in any case do not suffice, so that Chatelet’s result, even though it yields
some perfection at this point, is not improved in principle.
More precisely, we shall prove the following theorem:
Theorem 46.1. Let [A;: Q] < °°, and let V be a cubic surface of general
type over к with a finite number of (geometric) singular points and a dense
set of к-points V(k), while ^(КДС)) = {1}. Moreover, let there be given a finite
family of geometrically irreducible k-surfaces V) (i ~ 1,. . . , n) and k-mor-
phisms fi = Vf -> V of degree > 1. Пеп the set
n
of к-points of V which are not covered by this family is dense in the Zariski
topology.
The condition (7(C)) = {1} is automatically fulfilled if V is smooth, and,
probably, in the general case.
In order to apply this result to the Chatelet surfaces, we choose for the
birationally trivial surfaces. Then the degree of any morphism f^: V) -> V which
is surjective in the general point is greater than one, because the surface V is
birationally non-trivial: one has Br(K K)/Bx (k, К) = %2^ Z2. Because in
addition the number of R-equivalence classes is finite, it would follow from
the possibility of covering one class by a finite number of two-parameter fami-
lies of points that the whole surface could thus be covered, which contradicts
the theorem.
For general cubic surfaces, this result leads to the following alternative:
Corollary 46.2. Under the conditions of Theorem 46A,if V is birationally
non-trivial over k, then at least one of the following two assertions is true:
(i) The set E = V(k)/R is infinite.
(ii) No class X^E can be covered by a finite number of morphisms of
rational surfaces.
It is not impossible that both possibilities can be realized simultaneously.
Problem 46.3. Can every R-equivalence class be covered by a finite number
of morphisms of special varieties (of arbitrary dimension)?
46.4. Before turning to the proof of Theorem 46.1, we want to say a few
words as to its nature.
(i) . We explicitly construct ‘many’ points on V(k) which are not cov-
ered by a given family of morphisms. This construction is based on a method
of multiplying points on V(k) which has not been used up to now. Namely,
we consider a sheaf of hyperplane sections of which three base points are in
K(A:). If it is sufficiently general, almost all sections are elliptic curves. On
most of them the group of ^-points has infinite order: the triple of base points
already generates an infinite subgroup. The union of these sets of rational points
(for all sections), generated by the three base points, already shows that a cov-
ering by a finite number of morphisms is not possible.
(ii) . To prove the latter assertion, we have to use in an essential way that
the base field к is global; the theorem is not true if к = R or Qp because of ob-
vious topological reasons (see the proof of Theorem 16.1). We reduce the
problem to an analogous question for curves of genus 0 or 1 and use all
the known qualitative results on rational points on curves. These results show,
roughly speaking, that the density of rational points on curves falls sharply
as the genus increases (Lemma 46.13). Therefore, if we have a morphism of
curves /: Y X, while the genus of X is 0 or 1 and the genus of Y is greater
than the genus of X, then the set f(Y (kJ) is so much smaller than X(k) that
even a finite number of morphisms cannot manage to cover it.
This concludes our informal remarks and we proceed to the techniques
needed for the proof.
46.5. Survey of height theory. The height concept is the main tool in the
‘counting’ of fc-points on projective surfaces. It permits us to give a precise
meaning to the expressions ‘many’ and ‘few’ points and it will be essentially
used in the following. We shall omit most of the proofs, referring the reader
to Lang [1] or to Manin [2].
Let u be a place of the field k. It uniquely determines a norm
I I :&->R
и
which satisfies the following conditions:
(i) Ixly = I x I, where x is the image of x under the embedding in R or C,
if v is an Archimedean place.
(ii) I ply =р-1, if u is a non-Ar chimedean place corresponding to a divisor
of the prime number p G Z.
Further, let kv be the completion of к in the topology induced by u, and
let nv = : QJ.
46.6. Definition and Lemma. For every system (x0,..., xn) € of
elements of the field k, not all of which are equal to zero, the number
h(xQ,..., xn) = S nv max (log I x. l„)
(by definition, log 0 - is well defined and has the following properties'.
(i) й(Хх$,.. . , Xxw) = h(x§,. .., xn) for all X G k*. Therefore h can be
considered as a function on the к-points Pnfk) of projective space, once a
coordinate system has been chosen. This function is called the height.
(ii) й (x) > 0/or allx EPn(kf '
(iii) h (x) does not depend on the choice of the field to which the projec-
tive coordinates of x belong.
Example 46.7. Let к - Q, xy G Z, and g.c.d. (x0,.. ., xn) ~ 1. Then lxz-lv < 1
for every non-Archimedean place и and there exists an index i such that
I xju= 1, so that max;- (log Ix/ly) ~ 0. Consequently, in this case,
й(х0,..., xj = max log 1хуI.
In particular, the number of points in PW(Q) for which the height is bounded
by some constant, is finite. Mere generally:
Lemma 46.8. Let C> 0, d > 0, and d &Z; let Pn be a projective space with
a fixed coordinate system. Then the set
{x e PW(Q) I h(x) < C, [k(x) : Q] < d}
is finite.
The main inconvenience of this definition of height is the necessity of a
choice of a projective coordinate system. It turns out, however, that the height
only weakly depends on this choice. More precisely, let/p/2 be two real-valued
functions on some set. We shall call them equivalent and write Д if
1/1 ~ /2' *s bounded.
Lemma 46.9. Let hyth2be two heights on Pn(Q), defined in two different
•coordinate systems. Then 1ц ~Л2.
Henceforth, heights will be considered only up to equivalence; therefore
we shall be able to calculate them in any coordinate system. The reader can
convince himself that the truth of any assertion made does not depend on a
substitution of an equivalent height for the height considered.
We can now formulate the main general theorem on heights.
Theorem 46.10 (A. Weil). With each pair (K L) consisting of a projective
variety V over a field к and an invertible sheaf L on it, there corresponds a
height function 1ц : V(k) -> R. This correspondence satisfies the following
properties, which determine each height uniquely up to equivalence'.
®hLt®Lt~hLl+hL2-
(ii) If V = Pn, L = 0(1), then hg^ is the height defined in 46.6.
(iii) For every k-morphism : V -> W and L G Pic W we have h^ * ~hLo ip.
(iv) 1ц does not change under an extension of the base field.
We shall call the functions 1ц Weil heights. On Abelian varieties they have
an important additional property: they are quadratic with respect to the group
law.
Theorem 46.11 (J. Tate and A. Neron). Let V be an Abelian variety over к
and L an invertible sheaf on V. There exist uniquely determined functions
b^ : К(к) X K(k) R and l[ : И(к) -> R with the following properties:
(i) bL is a bilinear map, and lL is a homomorphism.
(ii) The Weil height h^is equivalent to the Tate-Neron height hL which is
defined by the formula
hL (x) = ^bL (x, x) + lL (x) .
(iii) If the sheaf L is very ample, then the quadratic form b^(x, x) on V(k)
is positive definite modulo torsion. Moreover, lj - 0 if i*L = L, where i(x) =
- —x, i: К К (symmetric sheaves),
(iv) If L is in the connected component Pic0 V, then bL =0.
For an elementary exposition in the case of elliptic curves see Cassels [3].
From this result and the functoriality of the Weil height we can derive
some properties of heights on curves which we shall need:
Lemma 46.12. Let V be a smooth projective curve over k, let be
two ample sheaves on V, and let deg Lj > 0 be their degrees. Then the Weil
heights (I/degLy)hL^ and (l/degLf) h^ are asymptotically equivalent on
V(kf, that is,
hL^(x) degZ1
*/,(*) ” de$ L2
0 for hL _(x) 00 .
Proof, (i). If the genus of Vis equal to zero and V(ty^ 0, then Vis isomorphic
to P1, and Li ~ Opi (1)^. The result therefore follows from Theorem 46.10(i).
(ii). Now let the genus of К be greater than zero,/^ the Jacobian variety of
the curve V, and $y : Vthe canonical embedding (it is determined up to a
translation; as before, we assume that V(k) is non-empty).
Replacing by L^Li, i Ф j, we can assume that degZq = degZ2 • Let
be a sheaf on J у such that *y(M j~ L , and ® M21 a sheaf on J у
which is in Pic0/^ and which induces L\ ® Z^on V. This sheaf exists be-
cause degZj = deg Z2 => ® £21 e Pic0 (K).
It follows from Theorem 46.11 that the quadratic parts of the heights h^
and hM coincide, and that these quadratic parts are non-degenerate (modulo
torsion^. Therefore the difference in the linear parts cannot affect the asymp-
totic equivalence of these heights. Because — hy^ о $v (Theorems 46.10
and 46.11), the lemma follows.
Lemma 46.13. Let V be a smooth projective curve of genus g such that
V(k) is non-empty, let L be an invertible sheaf on V of degree d>0. We de-
note by (H) the number of points x G V (k) for which hL(x) Then
the following estimates hold for H
(i) g = 0 : NL(H) = q exp (2ЯК) (1 + о (1));
(ii) g ~ 1 : ~ C2(H/d')rl2(\ + o(l)), where r is the rank of the group
V{kf
(iii) g > 1: Nl(H) < e3 log Я
Here the constants Cy c^ c3 depend on Vand к but not on L.
Proof (outline), (i). Suppose that к - Q and g = 0. Then V — P1. Example
46.7 shows that
^(Xq, *1) ^^inaxG°8 1, log I x* I) ,
where xQ, X| G Z are relatively prime. Therefore NjJH) coincides asymptoti-
cally with half the number of primitive points in the integral lattice in the
square (I x01, Ixj I) < exp (Hd~^). It is well known that this number is asymp-
totically equal to (12/тг2) exp (2Hd~^).
Schanuel [1] deals with the case of an arbitrary field к (and instead of
P1).
(ii) . Now let g = 1. Instead of hL we can consider hL. According to Theo-
rem 46.11, the quadratic part of hL has the form db$, where Z?o : 7(£) -> R is
positive definite and does not depend onZ. The natural map V(k) K(/c) R
has as its kernel a (finite) torsion group and the image is a lattice in the r-dimen-
sional Euclidean space V(k) R. The metric on it is defined by the form b§.
Therefore the region hL(x) < H in this space is a sphere of radius
(Hd~^ - Hq)?, the centre of which and Hq depend on the linear part of hL.
The number of points of the lattice in this sphere is proportional to its volume,
which establishes the desired asymptotic behaviour.
These arguments are due to Neron.
(iii) . Finally, let g > 1. The estimate for NL(H) in this case was obtained
by Mumford [1]. According to the Mordell conjecture, one must have7^(77)
< constant, but for our purposes the logarithm also suffices.
For Mumford’s proof one again considers a map of V(k) into a Euclidean
space J у {к) ® R. The metric in the latter is induced by the quadratic part Z>0
with respect to the standard ample sheaf 0(0) on Jy. The map
V(k) -* Jy(k) ® R is induced by <py\ its image is contained in a sublattice in
the image of Jy(kf and the inverse image of each point contains no more
elements than the order of the torsion of Jyfkf
Mumford’s basic result, which is obtained by means of a careful comparison
of the heights on a curve and its Jacobian using their functorial properties,
can be stated as follows (in a somewhat weakened form):
There exist two constants e > 0,77 > 0 such that for any two vectors
Jy(k) satisfying I (|| x || /|| у ||) — 11 < e, the angle between x and у
is greater than 77, provided that these points belong to the image of V(k).
In other words, vectors of almost the same length in V(k) cannot have al-
most the same direction. On the other hand, the collection of vectors in a
Euclidean space of given dimension r such that the angle between any two of
them is greater than 17 has no more elements than some constant TV, which
depends only on r and 77.
Consider now a set (at most countable) of points in Jy(k) ® R which
come from V, and arrange them in a sequence according to increasing length
Xp ... ,хй ... .It follows from what has been said that 11 xn+7V 11 >
> (1 + e) II xn II for all n. Thus the lengths II xn II do not grow slower than in geo-
metric progression. This immediately establishes the estimate (iii). The length
Z?0(x, x) and the height hL(x) differ by a factor d +o(l) and this can be taken
account of once and for all by enlarging the constant c3. This proves the lemma.
Lemma 46.13 permits us to establish a one-dimensional variant of the
main Theorem 46.1, to which we shall afterwards reduce this theorem.
In order to give an exact formulation we introduce the following definitions.
Let Ebe a curve over к and let the set E(&) be infinite. We shall say that al-
most all points of V(k) have some property P if
Л^(Я)
„lun NL(H) = 1 ’
where denotes the number of points which satisfy property P and the
condition hL(x)^H. Here L is some invertible sheaf of degree > 1. It imme-
diately follows from Lemma 46.12 that this definition does not depend on the
choice of L.
We can now formulate:
Corollary 46.14. Let f: V' -» V be a morphism of projective curves over k.
Suppose that the following conditions are satisfied:
(i) The curve V is geometrically irreducible and smooth, its genus is 0 or 1,
and the set V(k) is infinite.
(ii) For each irreducible component V- of V’ the induced morphism
ft'. P} -> V has degree > 1 and the genus of V- is greater than one if
the genus of V is equal to one.
Then almost all points of V(k) do not belong to the set /(F'(fc)).
Proof. It is obvious from the definition that if for every i = 1al-
most all points of И(&) have property then almost all points have all prop-
erties Pz simultaneously. It therefore suffices to consider the irreducible com-
ponents of V' separately, i.e., we can assume that Vf is irreducible over k.
If, moreover, У'®к is reducible, then the Appoints of V can only be at the
intersection of at least two conjugate geometric components; therefore there
are only a finite number of them, and the assertion is obvious .
Now let V' be geometrically irreducible, e > 1 the degree of the morphism
f. Then we have for every point x € V'(k) that
й/*(£)(х) ~, degf*(L) = edegL .
Therefore the number Tv£ (H) of points of L-heighth < H which belong to
/(/'(fc)) satisfies the estimate
л^(Я)<лу,(2.)(Я)(1 + о(1)).
Under the conditions of the lemma, this establishes the following inequalities,
where d = deg L:
(i) Genus Vis equal to zero:
Л^,(Я) = с,(ехр2Я(/-1)(1+о(1)))
X/ JL
zC^exp if genus V’ - 0 ,
TV? (77) < < c\ Hr^ if genus Vr = 1 ,
log H if genus Vr > 1 .
(ii) Genus Vis equal to one:
7У£(Я) = 4(Я£Г1)^(1+о(1)), r>0,
№l(H)<c'3 logH,
because genus V' is by hypothesis greater than one.
This proves the corollary.
Finally we shall use the estimates of 46.13 to study the specialisation of
the group of sections of a bundle of elliptic curves.
Let В = P1 over к, V a smooth projective surface over k.p : V В a k-
morphism of which the general fibre is geometrically irreducible and has
genus 1. Suppose further that the geometric fibres of V do not contain excep-
tional curves and that the ^-section : В -> V of the morphism p is fixed.
Then on an open set C V, the complement of the finite set of points
in which p is not smooth, there exists the structure of an Abelian scheme
over B. In particular, the sections K(B) of the morphism p form an Abelian
group with a finite number of generators and with as zero - 6$.
On the other hand, for each point x EB(k), except a finite number, the
fibre Vx is a one-dimensional Abelian variety, and the intersection with Vx
induces a natural homomorphism
px :F(B)->Fx(fc).
The following lemma, in a slightly less precise form, was first proved by
A. Neron.
Lemma 46.15. The kernel of the homomorphism px is trivial for almost
all x E B(k).
Proof. Let q > 1 denote an integer such that the order of the torsion group
in F(B) is relatively prime to q. The fib re wise multiplication by q is a B-mor-
phism q : Kq -» Vq (outside the points where p is not smooth).
Let Sq C Iq С V be the zero section. The inverse image q*(S§) C Kq is
closed in V and contains Sq as an irreducible component of multiplicity one.
Let </*(*$()) denote its complement. Further,let S),..., SN С V represent the
non-zero classes of V(B)]qV(B\ and let
N.
Z=1
(the addition is in the divisor group on K). All the curves q*(Sf) С Vare closed
in V and defined over k. We denote by f\ В’ В the restriction of p to B\ and
check that the conditions of Corollary 46.14 are fulfilled for this morphism.
The genus of В is equal to zero and the set B(k} is infinite. Moreover, none
of the irreducible components of the curve B* is a section. This is obvious for
the components of the curves z > 1, because ф qV(B), and for the
components of 7*(Sq) because the torsion of V(B) is relatively prime to q.
Consequently, according to 46.14, almost all points of B(k) do not belong
to/№)).
It therefore suffices to prove that if Ker [px : K(Z?) -> =# 0, then
there exists a point у such that x = f(y). Let px(S) = 0. If 5 qV(B),
then for some i > 1 and S’ G V(B) we have
S = P^-qp^S’)
(the addition is in the groups V(B) and Vx(k))t Therefore, on the curve z/*(5z)
there is a к-point over x: the intersection of S' with Vx.
We now consider the case S EqV(B). Let S = qaS’ and S’ not divisible by q
in V(B). Then either px(S') = 0, and the matter reduces to the previously con-
sidered case, or px(S’) Ф 0. In this case there exists an integer 1. < b < a
such that px(qb S’} = 0, px{qb~^S’) Ф 0. Set S” = qb~^S’. We have
<7Рх(Г) = 0, Px(S")*0.
This means that on the curve z?*(S0) there is a Appoint over x, the intersec-
tion of S" with an x-fibre, which does not belong to Sq. This concludes the
proof.
46.16. Proof of Theorem 46Д. We return, finally, to a cubic surface
V С P3 and a family of morphisms yj : -> V. Let Di C Fbe the branching
curve ofyf; it is non-empty because deg^- > 1 and ^(РДС)) = {1}.
In the space P3, dual to P3, we consider the set of those points for which
the corresponding planes contain no components of the curves Dt, no lines on
V, and no singular points on У. These points form a dense set. Because V(k)
is also dense in K, the set of lines in P3 which intersect V in three different
non-singular points of V(k) that do not belong to U or the lines of V,
is dense in the corresponding Grassmanian. Each of these lines determines a
bundle of planes in P3 passing through it, that is, a line in P3. Consequently,
in P3 there exists a Л-bundle of planes such that a general member of it has a
non-empty intersection with any component of the curves but does not
contain singular points of V; moreover, no plane of this bundle contains lines
on V, and the three base points of the sheaf of sections belong to
lines).
Perform a monoidal transformation with as centre the union of these three
points and resolve the singularities of F; let Vr be the resulting surface. Re-
place the morphisms^-: Vi V by morphisms f- : К/V\ where is some
resolution of the composed rational map V V'. As the assertion of
the theorem is of birational nature, it suffices to prove that the set
К'(Л) //(F-(£)) is dense in /(Л).
We consider the morphism p : V’ В of the surface V' into the base of the
bundle of plane sections. Again because of the birational nature of the prob-
lem we can assume that this morphism satisfies the conditions described pre-
ceding Lemma 46.15.
As the zero section in F'(S), we take the inverse image of one of the base
points of the bundle of sections of V, say x Then the section, the inverse
image of another base point y, has infinite order in the group K'(2?). In fact,
let z G К(Л) be a point on one of the smooth irreducible sections of the
bundle. The result of the composition of z and у with the zero x on this section is
written as tx ty(z) in our standard notation. Consequently, translating with
the inverse image of у in the group of sections V1 (B) induces a birational map
of V' which corresponds to txty on E But in 39.8.3 it has been proved that
tx ty has infinite order, in any case when V is non-singular. The same method
can be used to prove this result also for singular surfaces V; we omit the de-
tails.
It follows in particular from this that the group V'fB) is infinite, and in
virtue of Lemma 46.15 that the group Vx(k) is infinite for almost all x EB(k).
We denote by the normalisation of the curve В in the field of rational
functions on V- and consider the commutative diagram
We distinguish two cases.
(i) deg g{ > 1. Then according to Corollary 46.14 almost all points of
B(k) do not belong to g^B^k)). This means that on almost all fibres Vfx
there are in general no points from the set f-(V-(k)), although there are in-
finitely many fc-points on each of these fibres.
(ii) deg g; = 1. Then according to Bertini’s theorem all except a finite num-
ber of the fibres (Т/)Л7 x &B(k), are geometrically irreducible smooth curves.
Moreover, the genus of these fibres is greater than 1, because all the Vx have
a zero-dimensional non-empty intersection with the branching curves D- of
the morphisms//on V*. Applying Corollary 46.14 to the morphisms
(Vj)x Kx >we obtain that on those fibres 7X for which Vx(k) is infinite, almost
all points on are not in the image of (K/)x(&).
Combining these results we obtain that in all cases almost all ^-points of
almost all fibres of the morphism p over x^B(k) do not belong to
Ц” i particular, they certainly form a dense set on Vf, This proves
the theorem.
Problem 46.17. Let h be the height on a cubic surface V C induced by
its ample embedding. What is the asymptotic behaviour of the number
N(H) e V(k} I h(x) <Я}?
How essentially does it depend on the minimality of У?
At first one would have to study this problem for surfaces over Q which
are obtained by a monoidal transformation with as centre six Q-points and
for S?=o Г? = 0.
47. Counter-examples to the Hasse principle
This section contains explicit calculations to illustrate the principles de-
scribed in 41.14 —41.22 of this chapter.
We start with an example of Mordell [1].
47.1, The surface V is given over Q by the equation
3
Т3^ To +dr T3)(a2 To +d2 T3) = П(Г + 7\ + (0(Z))2 T2) .
r=l
Here ait di are integers, К = Q(O^) is an Abelian cubic extension which is
ramified in one place p = 7 or 13. For each value of p there are defined two
series of congruences for aif df. one of them ensures that K(Q^) is non-empty
for each q, the other ensures that the local components of the Brauer group
fit. (It is obvious that F(R) is non-empty, and the contribution of the real
places to the collection of local invariants is zero, because Br(K,£) consists of
elements of order 3.)
These series of congruences (see (А), (В), (C), (D) below) have infinitely
many solutions. Mordell constructs an additional sequence of examples, cor-
responding to a field К which is ramified in p = 3; we omit it because it
necessitates some more minute calculations. The essential picture is complete-
ly the same. Moreover, we restrict ourselves to an indication of the part of the
congruences which are sufficient to demonstrate the principle; the other suf-
ficient conditions can be found in Mordell’s paper.
Let us now describe all data successively.
47.2. The field К and 0. Let be a primitive root of unity of degree p.
К is uniquely determined as the cubic sub field of Q(f7) or Q(f 13). One of the
values of 0 is given respectively by the formulae
r+r1 +1 ,
for p = 7 ,
e =
4+f5+f8+f12-4, forp=13.
The number of classes of К is equal to 1; the ring of integers coincides with
Z[0]. The discriminant of 0 is equal to p2.
In the field К the number p ramifies as p = p 3; the prime numbers q for
which -
'± 1 (mod7) forp = 7 ,
v± 15 ± 5 (mod 13) forp=13,
decompose completely. The remaining q stay prime.
47.3. The Brauer group of V, We restrict ourselves to a consideration of
two elements of Br (Vr К). The divisors of the rational functions on V
T3 ' T3
belong to Л^д (Div V ® K). As in the examples already analyzed, they deter-
mine the elements of the Brauer group of V9 which we shall also denote by Д
abusing brevity.
The local invariants inv^ ()J(x)) for x Supp (/}) are represented by the
classes ffa) (mod N%-(A^-)), where q divides q in K. One can restrict one-
self to points x Ц Supp (/}) because inv$ (/Дх)) is locally constant in the
^-adic topology of F(Q$).
We shall now show how to ensure that the local invariants fit together.
This is an idea of Swinnerton-Dyer [1].
Proposition 47.4. Suppose that the sets V(Qq) for all primes q are non-
empty and that the following conditions are satisfied'.
(A) 0 £ a^ a2 (mod pf 0 £ dy — d2 — 1 (mod p), p - 7 or 13.
(B) g.c.d. (<Zp d2} consists of primes which completely decompose in K.
Then for all q Ф pf x G K(Qp), the local invariants inv^ ()J(x))? i ~ 1,2, are
zero, For q =p there does not exist a point x G K(Q^) for which simulta-
neously
inv (/j W) = inv (/2(x)) = 0 .
p x p
Therefore the set V(Q) is empty.
Proof. Let x - (r0, fp t2, t3) G be a point outside U Supp (/}), with
coordinates which are relatively prime. We consider three cases separately.
(i) q completely decomposes in K. Then K- = and inv^(/’.(x)) = 0.
(ii) q remains prime in K. Then 7V(A^) ={x G Q* I vq(x) = 0 (mod 3)}.
If the equality
holds, then it follows already from this that Д (x) and f2(x) aiv norms. Sup-
pose that this equality does not hold; then v- (Го + t^O + t20^) > 0 and hence
vq (*)) > 0 for z - 0, 1,2. Considering again the left-hand part of the equa-
tion of 7, we obtain ^q{dxd2t3) = 0. From the fact that t$, q, t2, t3 are
relatively prime it follows that this is only possible if, say, v^df) > 0. Besides,
in virtue of (B), vq(d2) = i^(r3) - 0. Hence
Ч/М'Мо +dlt3^a2to + </2r3^ = u<?(alZ0 +^i^3) = 0(mod3) ,
so that again/| (х),/2(х) are norms.
(iii) . q-p (i.e., 7 or 13). Then p = p3 and (0) = 1.
We show first of all that
+^lZ3^fl2r0 +^2f3^ = ^'
In the opposite case, v- (z0 + / j 0 + f202) > 0, from which vp(tQ) > 0 and
that means мр(с1^21з) > 0. By (A), ^(dj J2) ~ 0, so that vp(t3) > 0, from
which vp (П(Zq + Zj0 + r202)) Consequently. v^ (z0 + 6 + r202) 3 and
vp (*i) > 0, vp (r2) > 0, which contradicts the fact that Zo, t^ t2 are relatively
prime. Thus vp(fy(x)) = vp(f2(x)) ~ 0. Moreover,/j(x)-/2(x) = ^1 “^2 ” 1
(mod p) by condition (A).
From this it follows already that/^(x) and /2(x) cannot simultaneously be
norms in K- because the norm of a unit has the form ± 1 (mod 7) (for p-1)
and ± 1, ± 5 (mod 13) (for p - 13).
This concludes the proof.
Lemma 47.5 (Lemma on local solvability).
(i) q-p. The set F(Q„) is non-empty if
г
— a2 = d^ = 1 (mod 7)'
(/2 = 2 (mod 7)J
forp = 1,
^=^ = 1 (mod 13)\
J1=4(modl3)^ for p~ 13.
d~^ 5 (mod 13)J
(ii) q = 2 or 3. The set K(Q„) is non-empty if
4.
dy = 0 (mod qf d^ Ф 0 (mod q} .
(iii) q > 3, q Ф p. The set K(Q ) is always non-empty.
4
(C)
(D)
Proof. In each case we give a non-singular Z/(<?) point on the reduction of
the surface K(mod q). Well-known elementary arguments then permit us to
show that it lifts to a #-adic point. (The lifting must be done by steps from
mod qn to mod to find the correction for each step, one obtains a non-
homogeneous linear equation; as the coefficients of this equation serve the
partial derivatives of the left-hand part F of the equation of V in the lifted
point (mod q). Because this point is simple, not all of these coefficients are
zero.)
(i) q~p. Because Up (0) > 0, the reduced equation of V has the form
F = едТо +rfjT3)(i2 TQ +ВД - 73 = 0 .
A verification using conditions (C) shows that x = (1, 0, 0, — 1) lies on V
(mod q). It is non-singular because (BF/dT3) (x) Ф 0.
(ii) q = 2, 3. Conditions (D) show that the point x = (0, 0, 0,1) lies on
V (mod q). It is non-singular because (dF/ЭГ0) (x) Ф 0.
(iii) q > 3,q Фр. If q decomposes completely in X, then the right-hand
‘norm type’ part of the equation of V decomposes in into three linear
factors. Therefore on V there are nine lines which are rational over . If q
remains prime in K, however, then K(mod q) has a non-trivial Z/(^)-pomt x
by the classical theorem of Chevalley. If it is singular and double, then a line
through it in any Z/(t?)-rational direction intersects V (mod q) in a non-singular
Z/(t?)-point. Finally, a simple calculation, which we omit, shows that triple
points on V(mod q) do not exist.
47.6. To conclude, we describe the example of Cassels and Guy [1].
The surface V is given by the equation over Q,
V: 5T3 + 12T3+9T3 + 10T3 = 0.
That all the K(Q^) are non-empty is verified without trouble. In fact, if q > 5
is prime, then the surface V(mod q) is non-singular and it is well known that
it has Z/(#)-points, because the number of such points = 1 (mod q). On V
(mod 2) there is a simple point (1,0, 1,0), on V (mod 5) there is a simple point
(0, 2, 1,0). Finally there exists a 3-adic solution with To = 71| = l,T2 = O.
Let f be a primitive cubic root of unity. Because к = Q(f) is a quadratic ex-
tension of Q, K(Q) is empty if and only if V(k) is empty. We shall consider
к as the base field because the auxiliary fields к= &(^30) and = k($90)
are normal over k, Cassels and Guy work with Q instead of k, but all the for
us necessary number-theoretical information from their paper is easily carried
over to our base field.
The Galois group G = Gal (K/k) is generated by the automorphisms of
order threej s and 7: s($30) = f^30; ?(^90) - f^90;s acts trivially on k2,
and t on k\. The obstruction which Cassels and Guy construct lies in the
group of divisor classes of K. It is supplied by the function (cf. Section 41.18)
Tn + Г.(^/90)2/15
f=-------4---------EK(V).
r2 + r3(^30)/3
The investigations fall into two parts: a local and a global part.
The local investigation. Let x = (xq) be some Q-adele of the surface V. We
shall say that the function f is defined in x if,- firstly xq Supp (/), and,
secondly, vq (f(Xq)) = 0 for all q except for a finite number, where q is any
divisor of q in K.
Lemma 47.7. Let x be a Q-adble of the surface V in which f is defined.
Then the K-divisor
has the form
(/(%)) = 325-M+w%-(1+s+s2) ,
where c is some K-divisor, and 3 and 5 are K-divisors defined by the equations
З3 = (3), 53 = (5).
This is a straightforward reformulation of Lemma 1 of the paper of Cassels
and Guy. A proof (which there is also omitted as a matter of fact) is obtained
by an immediate calculation of exponents if one rewrites the original equation
of V in the form
5(TQ + 7,1(^90)2/15)1+r+f2 = - 9(T2 + r3(^30)/3)1+s+s2
The global investigation.
Lemma 47.8. (i) 3 is a principal divisor and 5 is a non-principal divisor of
the field K.
(ii) The class of the divisor c belongs to the subgroup of the class group which
is generated by the class of 5.
Cassels and Guy prove that the group of classes of the field
^o=Q(^3O,^9O)С К is isomorphic to Z3 and that it is generated by the
class of 5. The class of (/(%)) belongs to this subgroup, however, because f
even belongs to Kq(v) and not only to K(v). Since К is a quadratic extension
of Kq, 5 cannot become a principal divisor in A?: the class of 5 has exponent 3.
Corollary 47.9. V has no Q-points.
Proof. As a matter of fact, V does not even have Q-adelesx in which/is
defined and the divisor (f(x)) is principal in K. In fact, Lemmas47.7 and 47.8
together show that the class of the divisor (/(*)) coincides with the class of
5 -1, because s and r, acting trivially on the class of 5, also act trivially on
the class of c, so that дЛ+s+s2 js a principal divisor.
47.10. Discussion, The author would like to connect this very choice of
the function/, of which the divisor of the values is investigated, with the
Brauer group of the surface V. This can be done with some stretching.
We extend the base field k to k' = k($36) and construct the fields k[ = k'kt,
K’ = k' K, We write the equation for V k’ in the form
1ST* + (<^§6 T} )3 + (ЗГ2)3 + 30Г3 = 0 ,
and we consider the two rational functions
т2 +Г (^/3O)/3
r2 = -T=----------^'(v).
У36ТХ + 31\
Then, on the one hand, f-j\//2; on the other hand, as in Example 45.3:
(A)€7Vk'a; (DivK®r),
Therefore Д and/2 define elements of the Brauer group of V (although over
different fields k\ and k’o, respectively). Thus Lemma 47.7 can be interpreted
as information on the connection between the local invariants of these two
elements on Q-adeles of the surface V.
It can be clearly seen that the proof of Cassels and Guy is much too short
for the Procrustean bed of our general theory: the field k’ is only needed for
the stretching of the limbs....
48. Bibliographical remarks
The Brauer—Grothendieck group for schemes was introduced and investi-
gated by Grothendieck [2]. Its connection with the Safarevic—Tate group (in a
different context) was explained in the same paper of Tate where the theorem
mentioned in Section 31 was proved. Brauer equivalence was defined in Manin
(9]; the contents of Section 46 have been taken from Manin [5]. The cal-
culations in Section 45 and the construction of the general obstruction to
the Hasse principle are here published for the first time. A detailed survey of
analogous problems in the theory of elliptic curves has been written by
Cassels [3]; there one also finds an extensive list of references.
APPENDIX
ALGEBRAIC VARIETIES CLOSE TO THE RATIONAL ONES.
ALGEBRA, GEOMETRY, ARITHMETIC
Introduction
Let X be an absolutely irreducible algebraic variety over a perfect field k,
dim X = n. Recall three rationality properties of X.
(a) Xis called ^-rational, or A?birationally trivial, if a birational map Pnk-^Xexists.
(b) X is said to be stably ^-rational, if a birational map P™ x Xexists for
some m s* 0.
(с) X is called ^-unirational, if a rational map of finite degree Pnk~* X exists.
When the mention of the ground field к is omitted, it is understood that the
corresponding property is valid over к.
One easily sees that (а) Ф (b) Ф (c). (To prove the last implication, choose in P™*n
an n-dimensional ^-rational subvariety Y such that the composition У-> P™+n-^P™ x
X is generically surjective. If к is infinite one can find a linear subspace Y with this
property.)
The somewhat loosely defined class of ‘varieties, close to the rational ones’ includes
k-unirational varieties but is not exhausted by them. For example, let X-> Y be a
surjective rational map such that the base Y and the generic fibre Xv are close to
rational. Then X itself is close to rational. Fano’s varieties (see Section 4 below) also
belong to this class.
Smooth cubic hypersurfaces, moduli spaces of curves of genus < 13, linear algebraic
groups and their homogeneous spaces are rational or unirational.
The classical problem of existence of unirational, but irrational varieties was solved in
1971, and a strengthening of this result, the existence of stably rational but irrational
varieties was proved only in 1983. A brief exposition of this remarkable achievement
is given in Section 4.
The irrationality proofs are partly based on the principles which were studied in
“ Cubic Forms". Section 1 of this Appendix is devoted to the further connections
between the Galois cohomology of the Picard group (or the dual torus) and the
birational properties of the varieties, exploited in the simplest situations in Chapters IV
and VI. Section 2 introduces the technique of descent due to Colliot-Thelene and Sansuc
which is the far-going generalization of the methods of Chapter VI, in particular of the
Chatelet surfaces theory. In Section 3 some new results on the geometry of rational
surfaces are expounded. This may be considered as a natural continuation of Chapter V
Finally, Sections 5 and 6 are devoted to the questions about R-equivalence and
rational equivalence in connection with constructions of Chapter I. In particular, new
information about the structure of abstract CML’s is discussed.
Unfortunately many interesting results not directly related to “Cubic Forms” had to
be passed over in silence. We refer the reader in particular to the recent survey articles
by VP. Platonov [81], and VP. Platonov and A.S. Rapincuk [82] for information on the
arithmetic of linear algebraic groups.
1. Galois cohomology, Picard groups and birational geometry
1. Permutation modules. Let G be a profinite group. A continuous G-module P with
discrete topology is called a permutation module if it admits a Z-free base on which G
acts by permutations (in IV.29 such modules are called trivial). Two G-modules M, N
are called similar if M Ф P — M ® Q for appropriate permutation modules P, Q.
Denote by [M] the similarity class of M. Since H\G, P) - 0 if P is a permutation
module, Hl(G, M) depends only on [A/].
VE. Voskresenskii [101], using Hironaka’s resolution, has proved the following
generalization of Lemma IV.29.1.2. Let к be a field of characteristic zero, G = Gal к Ik,
X an absolutely irreducible variety over к, X = X ® k. There is an evident action of G
on Pic X
2. Theorem. If X, Y are projective smooth к-varieties, then their birational equivalence
implies [Pic X] = [Pic У].
3. Corollary, (a) For any field extension КЭ k, Hl(X, Pic X) is a k-hirational
invariant of a smooth projective variety X.
(b) If X is k-birationally trivial, then H\K, Pic X) = 0 for all К Э к.
4. Semigroup of similarity classes. If a smooth projective variety X is rational (i.e., X
is birational to P$), the G-module Pic X is Z-free of finite rank.
The following group-theoretic definitions fit well with this situation. It is tacitly
assumed that all G-modules in these definitions are Z-free of finite rank.
Denote by the semigroup of similarity classes of such G-modules with respect to
direct sum.
If G is finite, Hl(G, M) denotes the Tate cohomology, defined for all i 6 Z and
which satisfies the duality law Hl(G, Mj* = H~l(G, M°), where A* = Hom(A, Q/Z),
~ Hom(Af, Z). The group H~l(G, M) can be defined also for profinite G as
lim H~\GIH, Mh), where H C G runs over all open subgroups.
A G-module M is called flabby, if H~1(G>, M) = 0 for all open subgroups G’ C G.
The significance of this notion is explained by the following result due to Voskresenskil
[105]. Let Г be a k-torus, X its smooth projective compactification, D the group of
divisors of X supported at infinity X > T. Denote by T the character module of T. Then
(a) Pic У is a flabby Gal £/&-module.
(b) The module T has a resolution
O-^f^D^PicX-^O,
where D is a permutation module and Pic X is flabby.
Axiomatizing this remark, call any exact sequence 0—> P~+ F^>0 a flabby
resolution of M, if P is permutation and F flabby. Denote by FG C SG the subsemigroup
of similarity classes of flabby modules and set FG - {[&f°]| M flabby}. Finally, call a
G-module M invertible, if it is a direct summand of a permutation module. Let UG =
Fg Ci Fg be the subgroup of classes of invertible modules.
5. Proposition ([26]). (a) Each G-module M admits a flabby resolution
0-> P“>F-^0.
(b) The class [F] depends only on [М]. Let
p(M) = p([M]) = [F], <r(M) - a([M]) = p(M°)Q .
(c) The map p (resp. o) induces a surjection SG -> FG (resp. SG —> FQGf The restriction
of p (resp, cr) to Fg (resp, FG) induces mutually inverse isomorphisms FG -> FG
(resp. FG-~>FG). The restriction of p and a to UG induces -id.
(d) p(M) - p(N), iff there exist two exact sequences 0-> E^> P—> 0,
0-> V—> E~*R—>0 where P, R are permutation modules.
The structure of the semigroup SG and its filtration was studied in the papers [43],
[70], [16]. We mention the following facts. If G is finite, then FG is a group (coinciding
with UG), iff G is metacyclic. For finite G, UG is an abelian group of finite type which
may be infinite.
Applications to birational classification. For a smooth variety X over k denote by
I(X) the group Г(О|) /к*. Let KI к be a Galois extension with group G, XK ~ Xk ® K.
The varieties X, Y are said to be stably birationally equivalent over к, if there exists
a birational map X x Pk^> Y x Pb for a certain a, b.
6. Theorem. For X as above, let U EX be a dense Zariski open subset with Pic Uk ~ 0.
Then:
(a) The invariant p(X) ~ p(I(UK)) E FG depends only on the stable birational
equivalene class of X over k. It vanishes for birationally trivial X,
(b) IfX is a k-torus split over K, then
P(T) = p(T) = [Pic XK],
where Xis a smooth compactification of T.
(c) Let Z(K! k) be the semigroup of classes ofk-tori, split over K, with respect to stable
k-birational equivalence. Then p induces an isomorphism Z(Klk)^ FG,
Part (c) was proved by V.E. Voskresenskii [103]. Parts (a) and (b), slightly
generalizing Voskresenskii’s results, were obtained by Colliot-Thelene and Sansuc [26].
7. Invariant field of a cyclic group. Let p be a prime, К = Q(xx,..., xp ),and let G be
the group of cyclic permutations of (jq,..., xp). Swan [91] has found a necessary
condition for KG to be a pure extension of Q. Using the technique described above,
Voskresenskii [102] proved that it is also sufficient:
8. Theorem. KG is a pure extension ofQ iff either p or -pis a norm of a number lying
ind(^i/p).
9. Linear groups. Let H be a connected linear algebraic group over a number field k.
Weak approximation is valid for H, if the image of H(k) in the adele group H(Ak) is
dense. The obstruction group is denoted by
A(H)~ H(Ak)/H(k).
Similarly, the obstructions to the Hasse principle for the class of principle homogene-
ous H-spaces (or H-torsors) lie in the Safarevic-Tate set
Ш(Я) = Кег Н\к,Н)^]1н\к„Н)
If H is semisimple and simply connected, A(H) = 0. If moreover H has no Es factor,
Ш(Я) = 0.
Set
{b€BrH| b splits over £}, Br^H-B^H/Br k.
10. Theorem. Let H be a connected linear algebraic group without Es factors, V its
smoth compactification over к, S the torus dual to Pic V. Then:
(a) A(H), Ш (H) are finite abelian groups, depending only on the stable k-birational
equivalence class ofH.
(b) There is an exact sequence
0-> H\k, Pic V)*-> Ш (Я)->0.
Moreover,
H\k, Pic V)* = BraV.
When H is a torus, this was proved by Voskresenskii [105]. He also investigated the
general situation [101]. The given statement was proved by Sansuc [84] who used the
technique of pairing with the Brauer group developed in Chapter VI of this book.
2. The Hasse principle and descent on rational varieties
1. The Hasse principle. The theory of the Brauer obstruction to the Hasse principle
introduced in Chapter VI of this book was considerably extended and supplemented by
a general descent technique in the papers by Colliot-Thelene and Sansuc. This section
is devoted to their results. Our exposition is based particularly on the papers [27] and
[83], [24], [25], [21]. Cf. also [29], where interesting conditional results are deduced
from one of SchinzeFs conjectures.
We recall, that for a class У of algebraic varieties over an algebraic number field
к the Hasse principle holds, if for any X G У with all X(kv) non-empty X(k) itself is
nonempty. The classical Minkowski-Hasse theorem states that for the class of
projective quadrics the Hasse principle holds. Another such class consists of the
Severi-Brauer varieties, i.e., £-forms of projective spaces. On the other hand, for cubic
curves (and abelian varieties of arbitrary dimension), cubic surfaces and conic bundles
over P\ the Hasse principle fails. A general method which sometimes allows to prove
that X(k) = 0 although X(kv) 0 for all и consists in checking that the Brauer
obstruction is non-zero. This means that for all adelic points (x„) G X(Ak) there exists
an element A G Br X such that inv„ A(xv) / 0 (see Chapter VI for details).
A quite general theorem proved by Birch [9] with the help of the circle method
means, roughly speaking, that complete intersections of large dimension verify the
strong Hasse principle. By definition, this means that if VQ(kv) # 0 for all v, where V° is
the set of smooth points of V, then V(k) / 0 for any projective model V of V. Here is the
exact statement of Birch’s theorem.
2. Theorem. For the class V(n; h, d) of varieties in Pq1 defined by h equations of
degree d and verifying the condition
n-l-dimVsi„g>/!(/i + l)(d-l)2<'"1) y,ing=V->V°, dim0=-l,
the strong Hasse principle holds.
It follows in particular that in the class of smooth complete intersections the Hasse
principle holds (a) for quadrics of dimension ^3 (n 5); (b) for intersections of two
quadrics of dimension MO (n 13); (c) for cubic hypersurfaces of dimension ^15
(n 5= 17).
As Sansuc [83] puts it, for each of these three classes one can define two numbers n0,
nt such that for n < n0 counterexamples to the Hasse principle are known, for n n0 the
Brauer obstruction vanishes, and for n nx, v / oo one has V(Qv) 0 so that the local
obstruction vanishes iff V(R) / 0. Namely, n0 = 0, пг = 5 for quadrics; n0 = 6, = 9 for
intersections of two quadrics; n0 = 5, пг - 10 for cubic hypersurfaces. Therefore one can
conjecture that for n < nQ the Brauer obstruction is the only obstruction for Hasse
principle and for n > n0 the Hasse principle holds. A remarkable recent theorem by
Heath-Brown [50] proves this for cubic hypersurfaces and n nA ~ 10. Below we shall
describe certain new results for intersections of two quadrics.
3. Descent. Suppose that for a variety X a family of dominant morphisms : Y^X
is given with the property X(k) = U . irfYfkf). To prove that X(k) = 0 it suffices then
to demonstrate that for each Yf there exists a v such that Yfkf) = 0 (and hence
Yfk) - 0). On the other hand, if X(k) is nonvoid and the У, are in a sense simpler than
X, say, ^-rational, we get a passable description of X(x). This is the general idea of
descent which goes back to Fermat, Mordell, Selmer (for elliptic curves) and which
works well also for Chatelet surfaces (see Chapter VI, Section 45).
Colliot-Thelene and Sansuc developed a systematic procedure to generate the
‘descent families’ (У,, starting from X. Under some natural conditions the following
can be proved:
(a) The obstruction to the existence of a /с-point on X corresponding to (У; , ^)
vanishes or not simultaneously with the Brauer obstruction.
(b) Brauer’s obstruction for Yi vanishes: there is no ‘second descent’.
One may hope that for a certain class of varieties X the members of the descent
family Yj verify the Hasse principle. Results of this kind are proved in [21], [32].
We shall now describe the Colliot-Thelene and Sansuc method, following [27].
4. Torsors. Let S be a fc-torus X a ^-scheme. A torsor (or, more precisely, an S-torsor
over X) is a A>scheme J together with a morphism : J—> X and an action of S' on J
along the fibers of p7. This action should make J a principal fibre space in the etale
topology of X.
Up to isomorphism a torsor J is defined by its class [J] G Нг(Х, S) (etale cohomol-
ogy). In particular, for X ~ Spec к we have Я1 (Spec k, S) ~ H*(k, S(k)) (Galois
cohomology).
For each a G H^k, S) we shall denote by the same letter its inverse image in
Я\Х, S), On the other hand, given a point x G X(k), a class [J] G Hl(X, S) can be
specialized to give [ Jx] G H1(k, S). The following simple fact is easy to verify:
[Jx] = 0 <5 xep/J(k)).
Let us now choose a torsor J and denote by 0j: X(k) -> H\k, S) the map x h-> [ Jx],
For each a G H\k, S) construct a torsorpa : Ja -> A in the class [ J] - a G H\X, S), In
view of the above observation,
ад= II Р.ш= II pm.
a&ImOj /в(Л)И0
Thus a choice of a A;-torus 5 and of a torsor J furnishes a descent family as defined in
Section 2.3. The following proposition shows that this family is finite in cases interesting
for arithmetic. Denote by ZQ(X) the group of 0-dimensional A?cycles on X and extend 6,
to a homomorphism 0j : Z0(JT)-> Нг(к, S) by setting 6}(х) ~ соге8ад/А.([7X]) for any
closed point x G X.
5. Proposition, (a) For proper X, cycles rationally equivalent to zero cycles lie in
KerOj,
(b) If, in addition, к is either local or finitely generated over the prime subfield, then
the image of0jinHfk,S)is finite.
6. Universal torsors. For proper, smooth, rational X there is a canonical choice of S,
Namely, set So = Pic X.
Moreover, among the S0-torsors over X there are distinguished ones called universal
by Colliot-Thelene and Sansuc. To define them, consider the map
X^VX^^Hom^o.Pic^),
such that the homomorphism ^([ J]) maps a character Л : So ® k~> Gm ® к onto the
Gm-torsor Л* [ J ® k] which we identify with an element of Pic X
7. Definition and Lemma, (a) An SQ-torsor J is called universal, //^([7]) = idPic^.
(b) The equivalence relation defined by Oj on Xfor a universal torsor Jis stronger then
for any other torsor.
8. Existence and uniqueness of universal torsors. Suppose that X(k) / 0. Then the
extension of G-modules (G = Gal k/k)
1 -+ A* k(X)* -> k(X)* /Р 1
is trivial. Following [25], we shall call the class of this extension the elementary
obstruction (to the existence of a Appoint). Its vanishing is related to the existence of
universal torsors.
9. Proposition (a) Let X be a proper smooth rational variety. A universal torsor for X
exists iff the elementary obstruction for X vanishes. For a number field k, an equivalent
condition is the vanishing of the Brauer obstruction corresponding to the group
В = Ker[res: Br X-> Ц Br X„ /Br k„].
V
(b) The set of universal torsors is either empty or a principal homogeneous space over
H\k, So).
10. Theorem. Let X be a smooth proper rational variety admitting an universal torsor
J. Then for any smooth k-compactification Jc we have:
(a) Jc is rational.
(b) Pic J( is a permutation G-module.
11. A local description of universal torsors. For X as in Sections 2.9 and 2.10, the
Zariski open subsets U С X with Pic U = 0 form a base of topology. Choose such a U
and set F ~ X U. Consider a torus Tdual to к[U]* Ik* and denote by M the
G-module of divisors X supported by F. The exact sequence of character modules
0”» A [U]* /к * -> MPic X—> 0 induces the exact sequence of A-tori
l-> Sq-^ M-* T—> 1. Thus Mbecomes a 50-torsor over T. Set G = Gal k/k.
Choose a G-splitting к[U]* Z- k[U]*lk* = t. If U(k.) 0, one can put trx(cl f) =
f/f(x) for x E U(k). Using this splitting we can define a G-homomorphism A[f]-^ A[U]
and hence a A-morphism : U—> T.
The 50-torsor defined earlier induces then an 50-torsor over U. One can prove that it
is a restriction on U of a unique (up to isomorphism) universal torsor J a on X, and that
all universal torsors are obtained in this way.
12. Theorem. Let X be a smooth proper rational variety over a number field к
admitting a universal torsor J, and let X{kv) = 0for all v. Then
(a) The obstruction for the existence of a к-point on X corresponding to the descent
family of universal torsors {pa : Ja~+ X\a E Hl{k, SQ), Ja {k) # 0} coincides with the
Brauer obstruction for X {i.e., they vanish simultaneously).
(b) The Brauer obstruction for Jca vanishes.
The second assertion follows from Theorem 10.
As was remarked in Section 2.3, if X{k) # 0 and the varieties Ja with Ja {k) # 0 are
A?rational, this theorem gives a description of X{k). Unfortunately, Ja may not be
A?rational even for a 3-dimensional intersection of two quadrics. E.g., let X be given in
Pr by the equations x2 + y2 + z2 - uv = x2 + 2y2 + t2 - (u - v){u - 2v) = 0. Then 2f(R)
consists of two connected components. Hence the same is true for JC(R) where J is an
universal torsor so that J€ is not R-rational.
13. Application to tori. Let now Tbe a /с-torus, X its smooth ^-compactification.
Since Pic f = 0, the local description of universal torsors over X given in Section 2.11 is
applicable to this case, with T- U. The A?torus M, dual to the divisor group of X
supported in X > Tis ^-rational, since it has a permutation character module. From the
exact sequence 1—>50-^M-»T-»lwe derive the exact sequence
M(k)-> Т(к)Л Hl(k, So)-+ 1. Splitting t-+ £[T]* Ik* with the help of identity 1 G
T(k) and applying the explicit construction we get the following results.
14. Theorem, (a) The decomposition T{k) = Щ pa {Ja {k)) described in Theorem 2.2
coincides with R-equivalence.
(b) The map д'. T(k)-> Hl(k, So) induces an isomorphism T{k)!R2^H\kf Sf).
Both groups are finite if к is finitely generated over the prime subfield. In characteristic 0
wehaveT{k)/R —X{k)/R.
15. Applications to the generalized Chatelet surfaces. Consider an affine surface
y2 ~ az2 = П'= г Pj(X), where a G k* (k*)\ and the P,.(Л) G £[A] are irreducible
pairwise coprime polynomials. The descent method applied to a smooth compactific-
ation X of this surface leads to the descent family consisting of the following subvarieties
inA^+1:
V(cl,...,cr):Q^Pi(k)=ci(ui-av^), i = l,...,r,
where G k*. In particular, if r ~ 2 and , P2 are quadratic polynomials, these descent
varieties are intersections of two quadrics in A5.
16. Proposition. Let T be a certain class of generalized Chatelet surf aces X, Ж the
corresponding class of descent varieties V(c .. . ,cr). If the strong Hasse principle holds
for IT, then the Brauer obstruction is the single obstruction to the Hasse principle in К
17. The Hasse principle for the intersection of the two quadrics. In [32] the following
results are established. Let V be an intersection of two quadrics in Pnk, n 5* 4, over a
number field k. Assume that V is absolutely irreducible and non-conical. Assume
furthermore that in the ^-pencil of quadratic forms containing V there are no k-rational
pairs of forms of rank 4.
18. Theorem. The strong Hasse principle holds in the class of intersections of two
quadrics with the described properties verifying one of the following additional con-
ditions:
(a) V contains a к-rational pair of lines,
(b) V contains a к-rational pair of singular points,
(с) V contains a smooth к-rational quadric surface.
From this theorem and Proposition 2.16 the following facts can be derived.
19. Theorem. In the class of smooth projective models of surfaces y2a? = P(A),
deg P = 3 or 4, P without multiple roots, the following is true:
(a) the Brauer obstruction to the Hasse principle is the unique one.
(b) IfX(k) is non-void, then the Brauer obstruction to the weak approximation is the
unique obstruction, and X(k) /R Aj(Af) (set of 0-cycle classes of degree 1).
In particular, if deg P = 3 and P is irreducible, then X(k) 0 and weak approximation
holds for X. If deg P = 4 and P is irreducible, then the Hasse principle holds for X and,
for X(k) 0, weak approximation holds also.
In these cases the universal torsors J with J(k) # 0 turn out to be ^-rational.
20. Example. Let к - Q, a - -1, P(A) = - A4 + n where n = 24r+tm >(),m = 1(2),
0 =s / 3. Checking the local solvability conditions for x2 + y2 + A4 = n, (x, y, A) G Q
and applying Theorem 2.18, we get that there are no solutions exactly in the following
cases:
(a) i - 0, m 7(8),
(b) i — 2, m = 3(4).
21. Example. The equation x2 + Ъу1 ~ (n - Л2)(Л2 - n +1), where n 0,1 is an
integer, is solvable in Q, iff n > 1, and n / 32fl+1(3Z? + 2), a, b E Z, a s* 0.
3. Geometry of rational surfaces. Complements
1. Del Pezzo surfaces and conic bundles. Let X be a smooth projective surface over a
field k. We recall (cf. Section 24, Ch. IV), that X is called a Del Pezzo surface, if the
anticanonical bundle is ample, i.e. X = Proj Г(0“=о IV*)• Then d = (Ilv • ~
dim — 1 is called the degree of X. We have 1 d 9 and if d = 9 then X = P2
(and Xis called a Severi-Brauer surface).
Xis called a conic bundle if there is a surjective morphism/: X-* C, where C is a
smooth curve of genus zero and the generic fiber of/is also smooth of genus zero.
In papers by the author and V. A. Iskovskih the Enriques classification theorem for
rational surfaces was modernized and made more precise. In [59] it was finally proved in
the following form which describes the minimal models of all k-birational classes.
2. Theorem. Every minimal smooth projective surf ace over к is isomorphic to a surface
of one of the following two families:
(a) Del Pezzo surfaces with Pic X Z.
(b) Conic bundles with Pic X—Z Ф Z.
Moreover, surfaces of the first family are necessarily minimal. Surfaces of the second
family are not minimal in the following case:
(b1)(^-nj=3,5,6or(nx-nj = 8,x^F1.
Some conic bundles are also Del Pezzo surfaces. Namely, this happend in the
following cases:
(^2) * Цх)= T 2 or 4 and Xhas two different representations as conic bundle.
(b3) (£lx • £lx) “ 8 and % is not isomorphic to FNfor N > 2.
Finally, there are no minimal rational surfaces with (Dx - Dx) - 7.
Now, let G be a group. A G-variety is a variety with an action of G on it. A
G-morphism between G-varieties is a morphism commuting with the action of G. The
notions of G-minimal models, G-conic bundles etc. are self-evident.
In a paper of the author in 1967 it was shown, that if G is finite abelian, the analogue
of Theorem 3.2 is true for rational G-surfaces. In [59] the condition of abelianness was
removed, and the following result was established.
3. Theorem. Let G be a finite group, X a minimal rational G-surface. Denote by P(X)
the subgroup of the Picard group generated by the classes of G-invariant divisors. Then
either P(X) — ZorZ®Z and X is correspondingly either a G-Del Pezzo surface, or a
G-conic bundle.
This result reduces the classical problem of the classification of finite subgroups of the
Cremona group (up to conjugacy) to the problem of the classification of biregular
automorphisms of a well defined class of surfaces.
If G is infinite Xmay contain no G-invariant ample linear system. Here is a classical
example of such a situation.
Consider a pencil of cubic curves on P2 with nine base points and smooth generic
curve. Denote by X the result of blowing up the base points. The projection X-> Pl
onto the base of the pencil has nine distinguished sections, images of the base points.
Taking one for origin and defining on the generic fiber of X the structure of elliptic
curve, we get the group Z4 * * * 8 of translations acting upon X. No ample linear system is
stable with respect to this group. X can be thought of as a ‘degenerate Del Pezzo surface
of degree zero’, cf. Section 3.6 below.
In [48] M. Gizatullin proved that under some conditions on the action of G this
example is typical and classified elliptic bundles with non-discrete automorphism group.
We shall describe his results.
4. The surfaces . In this and the next section к is algebraically closed of
characteristic 5=5.
Consider on Pf pencils of cubic curves containing the following pairs of curves:
Ai(y): {x2(x2 - x0)(x2 - yx0) = O,xox2 = 0}, у a parameter /0,1;
A2* {<^0^1 %2 0? -^0 ’
A3: {x0x2 + xf = 0, x^2 = 0};
A4: {x0x2 + xf = 0, XqXx = 0}.
Denote by И^-(у), W2, W3, W4 the result of blowing up the base subscheme of the
corresponding pencil. The surfaces are pairwise non-isomorphic except that
^(y ) forZ(0) = Z(y), where Z(0) = (02 - p + l)30~2(/3 - I)2. Over the
base these pencils have two degenerate fibers each.
The group acts on in the following way:
i > 2: (x0: хг : x2) (*o : *-*4 : rx2).
Roughly speaking, exhaust all surfaces having a поп-ample G-invariant linear
system and automorphisms which are not translations along fibers. More precisely, the
following statement is proved in [48].
5. Theorem. Let X be a rational G-minimal surface without an ample G-invariant
linear system. Assume however, that for each element gE G' there exists a non-principal
divisor Dg with (D2) > 0 and such that Dg is linearly equivalent to g(Dgf Then
(a) (^•DJ = 0,rkPicY = 10.
(b) For some m, defines an elliptic bundle p: X^P1 without exceptional Curves
in the fibers.
(c) If X is not isomorphic to one of the , then Aut X contains an abelian subgroup of
finite index inducing translations of the generic fiber of p.
Some interesting arithmetic results on a class of cubic surfaces, generalizing the
so-called Markoff surfaces were proved by El Huti [41]. He proves essentially that the
automorphism group acts on integer points with a finite number of orbits.
We now turn to generalized and degenerate Del Pezzo surfaces. Their geometry was
investigated by Timms [96] and Du Vai [40] after Schlaffli and Cayley. For modern
versions, see M. Demazure [39], J. Bruce and C.T.C. Wall [14], J.-I. Merindol [74], H.
Pinkham [80], E. Looienga [73], Y. Naruki, T. Urabe [79]. The paper [37] by D. Coray
and M. Tsfasman is devoted to the arithmetic of degenerate Del Pezzo surfaces.
6. Definition, (a) A generalized Del Pezzo surface is a proper smooth rational surface
X such that is almost ample i.e. the map X-* Xr = Proj(®“=0 T(O^")) is birational.
(b) A generalized Del Pezzo surface is called degenerate if X! is not smooth.
The structure of the nondegenerate Del Pezzo surfaces is described in detail in
Chapter IV. We recall one of the main results. Over fceach nondegenerate Del Pezzo
surface X of degree 9 is isomorphic to P2, and if it is of degree 8 it is isomorphic to
P1 x P1 or F{. For 2 d 9 there exists a morphism/: X~+ P2 blowing down r = 9 - d
mutually non-intersecting exceptional curves. Their images can be any system of
r = 9 - d points in general position. The latter condition means that no three points lie
on line, no six lie on a conic, and, for r = 8, no cubic curve goes through all points having
one of them as double point. (In Chapter IV the necessity of this condition was proved
for all r and sufficiency for r 5* 3. The proof of sufficiency for r = 1,2 can be found in
Demazure [39] and Iskovskih [59]).
Degenerate Del Pezzo surfaces can be characterized in the same way. However,
among the points to be blown up, infinitely near ones may exist. Assuming the ground
field to be algebraicly closed, we give the appropriate definition.
Let X = Xr -4 Хг_г -------> “> Xo ~ P2 be a sequence of monoidal transform-
ations with centers£ 1^, 1 < r<8.
7. Definition. The points (xz) are in almost general position if any of the following
equivalent conditions holds.
(a) No two points x(. lie on an exceptional curve no four points lie on a line in
P2; no seven points lie on a conic in P2.
(b) All points x(- lie on a smooth cubic curve in P2.
(Recall that a point x over a surface X is said to lie on a curve С С X, if it lies on the
proper inverse image of C.)
8. Theorem. Let X be a generalized Del Pezzo surface of degree d. Then 1 d 9 and
X is either isomorphic to P2, or to F29 or else to P1 x P1, or finally to a blow up of 9 - d
points in almost general position over P2. Any such system of points leads to a generalized
Del Pezzo surface,
9. The Picard group and root systems. The analysis of the structure of Pic X, endowed
with intersection index and canonical class, made in Chapter IV for Del Pezzo surfaces,
is literally the same for generalized Del Pezzo surfaces. The essential difference between
the degenerate and nondegenerate case lies in the position of the cone of effective
elements in Pic X. The surface X is a smooth model of the anticanonical surface X' and
the inverse images of singular points can be characterized by their classes in Pic X which
are very special effective elements of this group.
Let us give some more details. Denote by lQ E Pic X the class of a line on P2, by
,..., lr the classes of the inverse images of the blown up points. Set w = -3/0 +
10. Proposition. Set
R = {I e Pic X| I2 = -2, (I, w0) = 0},
Rcn(X) = {I E. R\ I is effective} ,
R(X)-Ref{(X)URef{(X);
RfX) ~ {I E 7?eff(X)| I is irreducible}.
If3 =£ r 8, then R(X) is a closed and symmetric part of the root system R, R{(X) is its
base. The intersection graph ofRfX) coincides with the (dual) intersection graph of the
desingularization of the anticanonical model X’.
We recall that a subset P C R of a root system R is called closed, if a, ft E P,
a + fiER^a-V /ЗЕР, and symmetric, if P = -P.
Now the problem of the combinatorial description of singularities of degenerate Del
Pezzo surfaces can be solved in two steps. First, one describes closed symmetric subsets
of root systems R. Second, one clarifies which of them have a geometric realization. The
first problem can be attacked by means of the following general result.
11. Proposition. Let R be a reduced indecomposable root system, (a19..., ar)its
base, a = пгах + • • • + nrar its maximal root. Then every maximal closed symmetric part
ofR is a transform by an element of the Weyl group of one of the following sets Rs, Sg.
(a) Let 1 «51r and n( = 1. Then the base of Rc is {aj /V i}. and Ri consists of all
positive linear combinations of these roots in R.
(b) Let 1 < i r and щ prime. Then the base of Si is {-d,a^ J^i) and Si consists of
all positive linear combinations m^ in R with mt s 0(n().
Using this description recursively, one can list also all non-maximal parts. The
resulting combinatorial object can be given by a table.
12. Theorem. Let 3 < r 8. Then for any closed symmetric part PERr there exists a
degenerate Del Pezzo surface of degree d~9~ r realizing P, with the following
exceptions:
r = 7,P = (A1)7;r = 8,P=(A1)s,(A1)7Or(AI)4®Z>4.
As a matter of fact, appropriate realizations can be found in a family of degenerate
Del Pezzo surfaces whose base points lie on a fixed elliptic curve. For further details one
may consult the articles quoted in Section 3.5.
Assume now that a degenerate Del Pezzo surface X is defined over a non-closed field
k. In view of Theorem 3.2, it is still birationally equivalent to either a nondegenerate
Del Pezzo surface, or to a conic bundle. Now, the arithmetic of these standard models is
known to a certain degree. Hence one approach to studying Xis to understand the
geometry of the corresponding transform depending on the Galois action on Pic X and
singularities of X'. This program has been realized by D. Coray and M. Tsfasman [37].
13. Birational automorphisms of rational surfaces. Certain sets of generators of Bir
were determined in the last century both for closed and non-closed k. In the second case
the description is fairly complex. In Chapter V of this book a modern version of the
method of ‘virtual linear systems with prescribed singularities’ is developed and applied
to minimal cubic surfaces X, giving a complete presentation of Bir X (in an earlier paper
by the author similar results were obtained also for minimal Del Pezzo surfaces of
degree 1 and 2).
V.A. Iskovskih ([63], [64]) extended these results to Del Pezzo surfaces of lower
degree. Below we shall describe a part of his results.
14. Cremona group. Let к be algebraically closed. We shall realize the two-
dimensional Cremona group as the group of birational automorphisms of the quadric
F0 = P1xPl.
Set A = PGL(2, k) x PGL(2, k) and denote by т: P1 x P1 -> P1 x P1 the involution
interchanging the factors. Then Aut Fo is A x {1, t} .
Let 7Г : P1 x P1 —> P1 be the first projection. Denote by В C Bir Fo the subgroup of
birational maps, compatible with тт. В is a semidirect product PGL(2, k(t)) x
PGL(2, k). Choose a point x G P1 x P1. It defines a birational map ex : Fo~» Fo which
blows up x and blows down the fiber through x. To normalize all constructions, we shall
fix bihomogeneous coordinates (m0, mJ, (v0, uj on Fo, andpointsx0 = (0,1) x
(0,1), y0 = (1,0) x (1,0) and set
= V'o : (“<» X (V<” Vl) (“»’ Ml) X ’
To: (u0, «,) x (v0, v0) x (u1( u0).
Then the following result is true.
15. Theorem, (a) Bir Fo is generated by rand B.
(b) All relations between т and В are generated by
(T0ex у )3 “ 1
and the relations between r and A.
On the other hand, Gizatullin [49] described all relations between the classical
generators of the Cremona group, i.e., the projective and quadratic transformations of
P2
We turn now to the case of non-closed k.
16. Theorem. Let X be a minimal conic bundle with (flx • flx) 0. Then Bir X
preserves the structure bundle and is a semidirect product Bir Xv x G, where G is a finite
group of automorphisms of the base and Bir Xv is the automorphism group of the generic
fiber.
This theorem is proved in [53]. In [59], [63], [64] Del Pezzo surfaces of degree d = 1,
2,3,4 with Pic X = Z ф Z are studied. For d == 1,2 a full presentation of Bir Xis given;
for d = 3,4 only generators are known. For example, if d = 1,2, Bir X is generated by a
subgroup preserving one of two conic bundles and by a classical involution (Bertini and
Geiser) interchanging the two existing bundles. For details see the original papers. ~
17. Structure of some groups related to cubic surfaces. In [67], [68] D. Kanevsky
investigated certain abstract groups defined by a presentation of the same type as Bir X
for a minimal cubic surface X. We shall describe below some of his results.
We shall call an abstract cubic a set S with a ternary relation L C S x 5 x 5, satisfying
the following axioms.
(a) L is invariant with respect to permutations of factors.
(b)If(x,y,z),(#,y,z')(ELandx/у, then z = z'.
The reflection group Gs of an abstract cubic S is generated by symbols tx, x E S subject
to the following relations:
t2x = 1 for all x E S ;
for all (x,y,z)e L.
The following result is proved in [67].
18. Theorem. Let S be given effectively and LC S x S x 5 be decidable. Then:
(a) The word problem in Gs is decidable.
(b) The conjugacy problem in Gs is decidable.
(c) Any element of finite order in Gs is conjugate to either tx or to txtyt2for appropriate
x,y,zES. (Here of course the decidability of L can be dropped).
The proof is based on a direct description of Gs as a limit of amalgamated sums.
In [68] it is established that S can be reconstructed from Gs if L contains no triples
(x, x, z). Moreover, under some additional assumptions it is proved that Aut Gs is
generated by Gs and permutations of 5 preserving L.
19. Unirationality and stable rationality. Let X be an absolutely irreducible algebraic
variety over k.
The so-called Zariski problem is the question whether a stably ^-rational X is
necessarily ^-rational. Recently a negative answer to this question was given [6]: (a) in
the class of rational surfaces over a nonclosed field fc; (b) in the class of three-
dimensional varieties over C. In this section we shall state some results of [6] for
surfaces. They refer to conic bundles. (Some unirationality constructions for Del Pezzo
surfaces can be found in Chapter IV, Section 29.)
20. Theorem. Let к be afield of characteristic /2, P E Xc[x] and irreducible separable
polynomial of degree 3 with discriminant aEk*^ (&*)2. Then the surface X given by the
affine equation
y2 - az2 = P(x)
satisfies the following property: P* x Xis birational to P5k.
The proof heavily uses the torsor technique. Using P and the Weil descent functor
from the field К = k[x]/P, the authors construct a k-rational three-dimensional torus 5
and 5-torsor X with the following properties:
(a) X admits a section. Therefore J is birational to X x 5 and X x P*.
(b) J can be realized as a Zariski-open subset of an intersection of two quadrics which
is birationally trivial.
This establishes that Xis stably rational. On the other hand, V. A. Iskovskih has
proved in several papers the following theorem on the birational non-triviality of conic
bundles.
21. Theorem. A к-minimal conic bundle with s^b degenerate geometric fibers is
birationally non-trivial over k.
In [53] this is proved for 5 8, in [54] for s - 5,6,7 and in [56] for 5 = 4.
It is not difficult to construct a minimal model for Xfrom Theorem 3.20 and to
establish that 5 = 4. Moreover, the Gal £/fc-module Pic X turns out to be stably
permutational, i.e., similar to the zero module. Thus, we have:
22. Corollary. There exists a stably к-rational, but not k-ratiorial surface, with stably
permutational module Pic X.
4. The Liiroth problem and the Zariski problem in dimension 5=3
1. Liiroth problem. In 1876 Liiroth proved that every unirational curve is rational. In
1894 Castelnuovo proved that every unirational surface over the complex field is
rational. (For non-closed base field this is hot true even for surfaces with Л-points, as the
example of minimal cubic surfaces shows.) The question whether all unirational
varieties of dimension ^3 over C are rational is called the Liiroth problem. The question
whether all stably rational varieties are rational, as we mentioned already, is called the
Zariski problem. Until recently the answer to it was unknown even over non-closed
fields.
It was long conjectured that in dimention d s* 3 the Liiroth conjecture has a negative
solution, even for cubic threefolds. A very important contribution to this problem is due
to G. Fano ([45], [46], [47]) who has been studying for four decades the birational
geometry of three-dimensional hypersurfaces and complete intersections of small
degree, later called after him the Fano varieties. Unfortunately, it was during these four
decades that the technique of Italian algebraic geometry ceased to suffice for adequate
treatment of the geometric problems accessible to the marvellous geometric intuition of
the founders.
The new algebraic geometry was built by the successive efforts of 0. Zariski, A.
Weil, J.-P. Serre and A. Grothendieck. Only after the complete reconstruction of
foundations was accomplished and the new cohomological methods were developed, the
neoclassical period began, when old problems could be attacked with new weapons.
At the beginning of the sixties the Liiroth problem was considered unsolved, contrary
to some claims in classical papers. A principle obstacle was the absence of an effective
invariant distinguishing unirational varieties from rational ones.
At the Tata Institute Algebraic Geometry Colloquium 1967, the author made a
report on the current status of the unirationality problem, where three approaches to the
proof of irrationality were discussed.
(a) The Fano method. It consists in studying of the group Bir X, or, more generally,
of the set of birational maps X^ X' with the technique of virtual linear systems. If this
set is 'small’, e.g., in the simplest case finite-dimensional, then Xis not rational.
(b) The method of intermediate motives. Let X be a three-dimensional variety, A3(X)
its middle Grothendieck’s motive. In a paper, published in 1968, the author noted that
blowing up a point on X does not change h\X) and blowing up a smooth curve adds
h\У) ® L’1, where L is the so called Tate motive. Hence in the group of classes of
3-motives modulo formal sums X ® L lies a birational invariant of X.
(c) The Brauer group method. Let A" be a smooth proper unirational variety over C.
Grothendieck noted that Br X is a birational invariant of X isomorphic to H2 3(X, Z)tors
(torsion subgroup). Hence Xcannot be rational unless H3(X, Z)tors ~ 0. (Unfortunate-
ly, for classical examples, e.g. cubic hypersurface, 3-torsion vanishes.)
In 1971-72 three independent papers appeared using all these three methods to
obtain various examples of unirational but irrational varieties. V. A. Iskovskih and Yu.I.
Manin [65] applied the Fano method to quartic threefolds X in P4 and showed that Bir X
is finite (B. Segre earlier proved that some smooth quartics are in fact unirational). C.
Clemens and Ph. Griffiths [17], considering intermediate Jacobians instead of motives,
proved that the intermediate Jacobian of the rational threefold must be either a
Jacobian or a product of Jacobians, and that for a cubic threefold this property does not
hold. (It was long known that smooth cubic threefolds are unirational; in fact, they were
classical candidates for the role of counterexamples to the Luroth problem.) M. Artin
and D. Mumford [2] constructed unirational conic bundles X over a two-dimensional
base space for which Z2 C H3(X, Z). Since H3(X, Z) = H3(X x Pn, Z), their examples
are not even stably rational.
Finally, in 1984 A. Beauville, J.-L. Colliot-Thelene, J.-J. Sansuc and H.P.F.
Swinnerton-Dyer [6] solved negatively the Zariski problem, constructing non-rational,
although stably rational three-dimensional varieties over C. To prove stable rationality
they utilize the torsor technique, essentially for surfaces over a rational function field.
To prove non-rationality they use the Clemens-Griffiths method.
Below we shall describe very briefly some recent results of three-dimensional
birational geometry. For further details the reader may consult reports by V. A.
Iskovskih [60], [61], A. Turin [98], [99], A. Beauville [5], S. Mukai and H. Umemura
[77].
We start with describing two classes of varieties, close to the rational ones, which
were most intensively studied. They are Fano’s varieties (analogous to the Del Pezzo
surfaces) and birational bundles with rational base and rational generic fiber (analogous
to conic bundles).
2. The Fano varieties of the first kind. A smooth projective threefold X whose
anticanonical bundle = (O^)-1 is ample and generates in Pic X a subgroup of finite
index r, is called a Fano variety of the first kind and index r.
A thorough study of the anticanonical system on X made after Fano by V. A.
Iskovskih (cf. also [1]) led to a detailed classification scheme for the Fano varieties. We
reproduce it below, using the following notation: d ~ r~3(fl^1 • r = index,
g = genus of (Пх • П^1), A1,2 — the Hodge number, dimension of the intermediate
Jacobian. In Beauville’s notation [5], the manifolds of index 2 and degree d form a class
Ad, and those of index 1 and degree d form Bd. Here is a list of all Fano varieties of the
first kind.
(a) Index 4. P3, d = 1, g = 33; rational.
(b) Index 3. Quadric Q С P4, d = 2, g = 28, h1'2 = 0; rational.
(c) Index 2. Here 1 d 5.
A p sextic hypersurfaces in the quasihomogeneous projective space P(l, 1,1,2,3); it
can also be realized as a double covering of a Veronese cone; g = 5, A1,2 = 21. A
‘general’ manifold is irrational (proof by the intermediate Jacobian method, A.
Beauville [4]). Unirationality is unknown.
A2: double covering of P3 ramified at a quartic, g = 9, A1’2 ~ 10; unirational;
‘generally’ rational (A. Beauville [4]); the Clemens-Griffiths method).
A3: cubic in P4; g ~ 13, A1’2 = 5; unirational; non-rational (C. Clemens, Ph, Griffiths
[5])-
A4: complete intersection of two quadrics in P5; g = 17, h1,2 = 2; rational.
A5: linear section of the Grassmanian G(2; 5) in Plucker’s embedding; g - 21,
h1 ’2 = 0, rational.
(d) Index 1.
B2: double covering of P3 ramified at a sextic surface; g ~ 2, A1,2 - 52; non-rational
(proved in [61] by Fano’s method); unirationality unknown.
B4: quartic in P4; g = 3, A1’2 = 30; irrational (proved in [65] by Fano’s method). Some
smooth quartics are unirational; for a general quartic unirationality is unknown.
B4. double covering of a quadric in P4, ramified at a surface of degree 8 (intersection
with a quartic); g ~ 3, h1,2 = 30; unirational; irrational (proved in [61] by Fano’s
method).
B6: complete intersection of a quadric and a cubic in P5; g ~ 4, A1’2 - 20; unirational;
irrational (proved in [61] by Fano’s method).
complete intersection of three quadrics in P6; g = 5, A1,2 = 14; unirational;
irrational (Beauville [4], the Clemens-Griffiths method).
B10: intersection of a quadric with G(2; 5); g = 6, A1,2 = 10, unirational; ‘generally’
irrational (A. Beauville [4], the Clemens-Griffiths method).
B12: a subvariety of P8 (we omit a description which is rather cumbersome); g = 7,
A1,2-5; rational.
B14: a linear section of the Grassmanian G(2, 6) in P14; g = 8, A1’2 = 5; unirational;
non-rational (in fact, it is birationally equivalent to a smoth cubic in P4, c.f. [61]).
Bd (d ~ 16,18,22): certain subvarieties in Pa/2+2; А1,2 = Ц - d!2, g = (d + 2)/2; all
rational, with the possible exception of some varieties B22. On B22 see also [77].
Finally, we should note that for many Fano varieties XV. A. Iskovskih described
geometric generators of Bir X. In some cases relations are also known.
3. Birational bundles. Let/ : X-> S be a morphism of a threefold to a rational
surface. If a generic fiber Xrj is also a rational curve, then birationality properties of X
depend first of all on the existence of rational section of/. If it exists, X is rational.
Otherwise it is a conic bundle, and birationality properties of X are to a great degree
determined by the properties of the discriminant curve С C S of/. One can choose a
model X in such a way that over a smooth point of C the fiber of/is a union of two smoth
rational curves, transversally intersecting at a common point. One can also arrrange that
C has only nodes, over which/has double P1 as fibers. Below we shall assume that a
conic bundle X is chosen in this way.
There is also the second natural class of birational bundles which are fibered by
rational surfaces over a rational curve / :X-^C. Here the generic fiber X^ has either a
model of Del Pezzo type over Cv = C(t), or a model which is a conic bundle over Cri. In
the latter case X itself evidently has also a conic bundle model.
The varieties of this class are in general not well understood. One may conjecture
that a conic bundle with a curve of a sufficiently large degree (over a given S) is not
unirational, but this remains unproved.
The following is known, however:
4. Theorem, (a) A conic bundle over P2 with a discriminant curve of degree 5=6 is not
rational, since its intermediate Jacobian is not a product of Jacobians of curves (see [5]).
(b) Let f : X^Sbe such a conic bundle with a discriminant curve C, that HQ(S,
(O5)40Os(C))#O. Then any birational automorphism ofXpreserves f, and X is
irrational (see V. Sarkisov [86]).
5. Stably rational irrational varieties. Let us consider now surfaces, as described in
Theorem 3.20, over a field к = C(z):
y2 - a(t)z2 = P(x, t),
where a(t) G C[t] is a polynomial without multiple roots which coincides with the
x-discriminant of an irreducible cubic polynomial in x, P(x, t) E C[x, /]. Denote by X a
model of C(x, y, z, t). This variety is stably rational in view of Theorem 3.20. On the .
other hand, one can find a conic bundle model of X and calculate its intermediate
Jacobian in terms of its discriminant curve. This gives the following result.
6. Theorem. If P(x, t)~0 is a curve of genus >2 whose projection onto the t-axis has
no points with ramification index 3, then the intermediate Jacobian of X is not a product
of Jacobians of curves. Hence Xis irrational.
7. Open questions. Although the last decade has seen considerable progress in our
understanding of the geometry of threefolds close to the rational ones, it is clear that
only fragments of a complex picture emerged as yet. The following questions arise
naturally:
(a) What can be said about the classification of Fano varieties of dimension 5=4? Is
their anticanonical degree bounded (by a function of dimension)?
(b) Find efficient criteria of irrationality in dimension s=4. Is a cubic fourfold
rational?
(c) The same as (b) for unirationality.
(d) Can a smooth rational variety be a flat deformation of an irrational one?
5. Rational points and equivalence relations
1. Bloch’s exact sequence. Let X be a rational proper smooth geometrically
irreducible variety over a perfect field k,G~ Gal k/k. Let A0(X) be the zero-cycle class
group of degree zero modulo rational equivalence. Denote by S the torus dual to
Pic X. If X admits a universal torsor J, the map 0j described in Section 2.4 induces a
homomorphism which does not depend on J,
Ф : A0(X)-> H\X, S)
(see Proposition 2.5). In tact, one can easily define Фdirectly without using/, cf. [28].
As in Proposition 2.5, Im 0is finite, if к is absolutely finitely generated. S. Bloch in
[13], using algebraic K-theory, has included (for dim X = 2) Ф into an exact sequence
giving important additional information.
SelF=k(XfF=k(X).
2. Theorem ([13]). //dim X ~ 2, there is an exact sequence
S(k)~*H\G, K2FIK2k)—> А0(^)Л H\k, S)-> H2(G,K2F/K2k).
Colliot-Thelene in [19], using recent deep results by Merkuriev and Suslin [75],
proved the following facts (statement (b) was known earlier).
3. Theorem. Let X be a rational surface.
(a) If к is a number field (or is finitely generated over Q and there is a к-cycle of degree
1 on X), thenA()(X) is finite. The same is true for a local field к; if furthermore X has a
nondegenerate reduction, then A Q(X) — 0.
(b) If char к И 2 and cd к - 1, then A 0(X) = 0.
(c) IfX is a conic bundle, char к / 2 and either X admits a Q-cycle of degree l,ork is
local or global, then Ker ф = 0.
Knowing qualitative properties of A0(X) one naturally tries to calculate it by local
means. As an example we shall again consider the Chatelet surfaces
y2 - dz2 ~ (x - Cj)(x - e2)(x - e3), d G k* (k*)2, ete7fori^j.
4. Theorem. IfX is a Chatelet surface, the following assertions hold:
(a)AQ(X)-X(k)/R.
(b) Every element of A 0 (A") is defined by local invariants: there is an exact sequence
O^/o(ZKll А0(Х,)->Н\к,$У
V
andH\k, 5) = (Z/2Z)2.
Cf. also Theorem 2.19 where a different class of conic bundles is treated.
The following explicit calculations can be found in [28]. We shall denote by Xrs the
Chatelet surface with parameters (d,et, e2, ef) - (~ 1,0, r, s) over Q. Then we have the
following cycle groups:
(',’) (1,5) (1,3) (3,7) (3,9) (7,19) (1,17)
A,VO Z/2Z (Z/2Z)2 (Z/2Z)3 (Z/2Z)’ (Z/2Z)4 Z/2Z
No surface in this table satisfies the weak approximation principle.
5. Tsfasman’s surfaces (cf. [37], [85]). There exist rational surfaces for which the
group
Ш A0(X) = Ker(X0(XHlI Aq(Xv))
V
does not vanish. Denote by Yab a smooth projective model of the surface
y2 ~ abz2 = (x2 ~ <z)(x2 - b).
Let k be a number field, К ~ k(Vd, Vb), G ~ Gal Klk, [X : k] ~ 4.
In this case UI Л 0CY) = Z/2Z iff there is no valuation v such that Gv = G; otherwise
шлдн
For example, if к = Q, we get a nonvanishing ПТ A 0 for («, b) = (p, q), where p,
mod 4 are primes such that (plq) = ~(p/q)Aqlp)4 = 1, e.g. (5,29), (13,17), (5 >
41), (13,29).
Tsfasman’s examples, e.g. У13Д7 have also the following interesting properties:
(a)Br У/BrQ^O.
(b) The Brauer equivalence on У is trivial.
(c) The equivalence on Yis non-trivial.
We now turn to the R-equivalence on intersections of quadrics. The following result
is stated in [32].
6. Theorem. Let XGPkbea smoth intersection of two quadrics over a number field k,
n^7, X(k) -A 0. Then the following assertions hold:
(а)ад/Т? = Щр=11770(Ж))-
(b) For archimedean v, X(kv )/R~ irQ(X(kv )fifor non-archimedean v, X(kv)/R = 0.
In a series of papers, D. Coray and coauthors have investigated the following
question. Following Colliot-Thelene. and Coray [20], we shall say, that a variety X over к
satisfies condition (A), if there exists a finite set of extension fields KJ к with relatively
prime degrees such that all X(KJ / 0. Obviously, (A) holds for X, if X(k) / 0. What can
be said about the opposite assertion?
7. Theorem, (a) From (A) it follows thatXfk) 0, ifX is a nondegenerate Del Pezzo
surface of degree s?4. This is true also for a smooth cubic X,ifkisa local field.
(b) Assume thatX is a smooth cubic surface satisfying condition (h). Then there exists
an extension field K/k with [X : k] = 1,4 or 10 such thatX(K) 0.
(c) Let X be a rational conic bundle, r =?= 8 - (flx • flx). If Xsatisfies (A) and
X(k) = 0, then X(K) ^fyfor a certain К of odd degree [X: k] < 5 - max(l, [r/2]), and
also X(L) t- fifor a certain quadratic extension field L.
In particular, for r =^5 from (h) it follows that X(k) 0.
However, for r ~ 6 there exist conic bundles over Qp, for which X(Qp) = 0 but
X(K) # 0 for appropriate extensions of degrees 2 and 3.
For proofs see D. Coray [34], D. Coray [35] and J.-L. Colliot-Thelene, and D. Coray
[20].
6. Cubic surfaces and commutative Moufang loops (CML)
1. Notation. Let X be a smooth cubic surface over a field K. We shall consider three
admissible equivalence relations on X(K): U (universal), R and Br (Brauer equivalence
related to the full group Br X). Recall that for any admissible equivalence relation S
there is on X(K) IS a structure of CML of period 6 (depending on a choice of identity
class). We shall denote by 52, S3 such equivalence relations that S ~ S2 A S3 and
X(K) I Sm is of period m.
In this section we shall describe certain results on the structure of X(K)/S obtained
since the first publication of this book, and also results on the structure of abstract
CML’s with finite number of generators, answering some questions raised in Chapter I.
We first note that the principal qualitative question on the structure of CML’s of
cubic hypersurfaces remains unsolved: do there exist non-associative CML’s of this
type? A key particular case of this situation is the following one. Let X be a generic
cubic surface over the field Q(a;yjt), where the aijk are independent coefficients of an
equation of X and let xt be four independent generic points of X. What CLM is
generated by xf over Q(ai/k)?
We turn now to finite fields. Almost exhaustive information in this case is supplied by
the following result of Swfnnerton-Dyer [95], which in a less precise form was
independently established by D. Kanevsky [66].
2. Theorem. Let X be a smooth cubic surface over a finite field к with q elements. Then
X(K) /U = 0, except for certain cases when X contains no к-lines and all points of X(k)
are Eckardtpoints. In the latter case we have either q = 2, card X(k) == 3, or q = 4,
card X(k) ~ 9.
The last case is realized for the surface x3 + y3 + z3 + Ot3 ~ 0, в2 + 0 +1 = 6,
considered in this book (Example 16.2). Swinnerton-Dyer proves that X(k) /R = 0 for
this surface, so that universal equivalence is strictly stronger than the R-equivalence.
Let now К be a local number field, к jts residue field. Assume that X is defined by an
equation F with integer coefficients such that the reduction F defines a smooth k-surface
X. Denote by H = | det(32F/3x/5x/) the Hessian of X. It has integer coefficients and
has a reduction H.
3. Theorem ([95]). Let X be a smooth cubic surface over a local number field К with a
smooth reduction. Then X(K) / U == 0 except for, possibly, the following cases:
(a) char к = 3 and either H^tfor the surface H = 0 touches X at all к-points ofX.
(b)char k = 2andH = ().
(c) char к - 2 and any rational point ofX is an Eckardt point.
4. Corollary. Let X be a smooth cubic surface over a number field K. Then for all v,
except for a finite number, X(KV) / U ~ 0.
The map X(K) —> Пи X(Kv )IUinduces on X(K) an admissible equivalence relation,
approximating U. To calculate it, we lack the understanding of the structure of
X(KV) / U for exceptional cases in Theorem 6.3 and for degenerate reductions. In order
to go from к to K, the following result of Kanevsky [69] can be used.
5. Theorem. Let X be a smooth cubic surface over a local number field К with possibly
nonsmooth reduction X. Assume that X(K) contains a point of general type and let
x E X(k) be a point in general position with certain pointy E X(k) (this means that the
line containing x and у doesnot touch X).
//char к 2,3, then all points {x E X(K)| x reduces to x} belong to the same
U-equivalence class. If char к = 2 (resp. 3), these points belong to the same class modulo
U3 (resp. U2) equivalence.
Finally, let L/K be a Galois extension with group G. An equivalence relation 5 on
X(L) is called G-invariant if the action of G preserves equivalence classes.
6. Theorem ([69]). If Kis infinite and there is a point of general type in X(L), then U
and R are G-invariant. The same holds for Br, ifX is smooth and char К = 0.
We shall now turn to the structure theory of abstract CML’s.
7. Nilpotence class. Question 10.3 on the nilpotence class k(n) of a free CML with n
generators was solved independently by L. Beneteau [7] and J.D.H. Smith [89]. They
proved that Bruck’s upper bound is in fact achieved:
8. Theorem. k(ri) = n - Iforall n^2.
9. Beneteau’s construction. The Beneteau proof is based on the following construc-
tion, which generalizes an idea of Bruck (Example 10.5).
Consider an associative ring R and a subset SCR with the following property: for
every triple x, у, z E S we have
x2 = xy + yx = 3xyz “ 0.
Denote by G the additive subgroup of R, generated by all products of an odd number of
elements of S. Define on the set E ™ G x G a multiplication law
CWaX-Vi, У 2) = (*1 + Л + - y2),x2 + y2+ x2y2(yt - x,)).
Then E is a CML.
Beneteau applies this construction to the ring R = ф^=0 AkL where L is an
infinite-dimensional linear space over F3. Take a base of L for 5; then G =
Фл=0 A2k+1L. In the CML E = G x G consider a subloop Tn generated by the
following elements:
i (Ji’ h+n)’ L • • • ? т
where Ц., ln is a base of L.
Beneteau proved that the nilpotence class of Tn is n - 1 for all n 2. In view of the
Bruck theorem, this proves Theorem 6.9.
10.3-order of a free CML. The second part of Question 10.3 on the 3-order l(n) of a
free CML E(n) with n generators is unsolved yet, but in a beautiful paper [90] J.D.H.
Smith formulated a conjecture on the value of l(n) and made some progress in proving
it. Let E(n) = Eq D Ex D • • Э En_x = {1} be the lower central series. Set 3(n) =
dimF3®"=1 £(/E(+1.
11. Smith’s conjecture. Set
D(z) = 2 5(w)z"/n!
n = 3
Then
D(z) = [ ze2z( J—“P2- - z2t(l - t) -1) dt
Jo V0(2z(r(l - Z))1/2) v 7 7
t2 I l-2?\ u z
+ res,_0 --? exp I z---r sh - .
1 - t2 F \ 1 - t2 > t
Here J^z) = S*=o (~1)“(z/2)27(k!)2 is Bessel’s function. Setting S^=o &х7(/<!)2 =
(J0(2Vx))-1 and «j = 0, aK = pK for к 2, we can represent this conjecture in an
equivalent form:
[л —1/2] п-2к-1
s(«)= 2 X
k=1 p = 0
р!(2к + 1)!(n -p “ 2k - 1)!
«к
/p + к - 1\
\ p /
n^3.
Here are the first few values of this function:
5(3) = 1, 5(4) = 8, 5(5) = 44, 5(6) = 214, 5(7) = 1000, 5(8) = 4592.
In effect, Smith proves the following group-theoretic fact. Let G be a finite group,
containing a product of two symmetric groups Sfc x Sk. Denote by 5y (resp. r;) for
1 j < к the permutation of £th and (к + 1 )th object in the first (resp. second) group
Sk. Set R = X[G] where К is a field. Then for к > 1 we have
к-1
coding 2 (1 + <ry)(l + tJR = |G| res2=0J0(2Vz)-1z-(4+1) = |G|&.
A conjectural part of Smith’s results relates this group-theoretical theorem to the
Фп —2
,_0 ^/Д+1 is considered as a
module over a group permuting the free base of E(ri). A conjecture on the structure of
this module is made which would follow from the ‘triple argument conjecture’ [89].
[1] A.B. Altman and S.L. Kleiman, Foundations of Fano schemes, Compos. Math. 34(1) (1977)
3-47.
[2] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not
rational, Proc. bond. Math. Soc. 25 (1972) 75-95.
[3] M.I. Basmakov and A.L. Cistov, On rationality of a certain class of tori, Trudy Steklov Inst.
48 (1978) 27-29 (in Russian).
[4] A. Beauville, Varietes de Prym et jacobiennes intermediaires, Ann. ENS10 (1977) 309-391.
[5] A. Beauville, Varietes rationnelles et unirationneles, Leet. Notes Math. 997 (Springer,
Berlin, 1983) 16-33.
[6] A. Beauville, J.-L. Colliot-Thelene, J.-J. Sansuc and P. Swinnerton-Dyer, Varietes stable-
ment rationnelles non rationnelles, Ann. of Math. 121 (1985) 283-318.
[7] L. Beneteau, Problemes de majorations dans les quasigroups distributifs et les groupes de
Fischer, Actes du Coll. Alg. Appl. et Comb., Grenoble (1978).
[8] L. Beneteau, Free commutative Moufang loops and anticommutative graded rings, J.
Algebra (1979).
[9] B.J. Birch, Forms in many variables, Proc. Royal Soc. London Ser. A 265 (1961) 245-263.
[10] B.J. Birch and H.P.F. Swinnerton-Dyer, The Hasse problem for rational surfaces, J. Reine
Angew. Math. 274(1975) 164-174.
[11] S. Bloch, Lectures on Algebraic Cycles, Duke University Math. Series 4 (1980).
[12] S, Bloch, Torsion algebraic cycles, K2, and the Brauer group of function fields, in: Groupe de
Brauer, Lecture Notes in Math. 844 (Springer, Berlin, 1981) 75-102.
[13] S. Bloch, On the Chow groups of certain rational surfaces, Ann. Sci. Ec. Norm. Sup. Ser. 414
(1981)41-59.
[14] J.W. Bruce and C.T.C. Wall, On the classification of cubic surfaces, J. Lond. Math. Soc. (2)
19(1979)245-256.
[15] A.L. Cistov, On the birational equivalence of tori with cyclic splitting field, Zapiski Nauc.
Sem. LOMI64 (1976) 153-158 (in Russian).
[16] A.L. Cistov, On the number of generators of the semigroup of classes of algebraic tori
modulo stable equivalence, Dokl. Akad. Nauk SSSR 242(5) (1978) 1027-1029 (in Russian).
[17] C. Clemens and Ph. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of
Math. 95 (1972) 281-356.
[18] J.-L. Colliot-Thelene, Surfaces de Del Pezzo de degre 6, C.R. Acad. Sci. Paris 275 A (1972)
109-111.
[19] J.L. Colliot-Thelene, Hilbert’s theorem 90 for K2, with application to the Chow groups of
rational surfaces, Invent. Math. 71 (1983) 1-20.
[20] J.-L. Colliot-Thelene and D. Coray, L’equivalence rationnelle sur les points fermes des
surfaces rationnelles fibrees en coniques, Compositio Math. 39 (1979) 301-332.
[21] J.-L. Colliot-Thelene, D. Coray and J.-J. Sansuc, Descente et principe de Hasse pour
certaines varietes rationnelles, J. Reine Angew. Math. 320 (1980) 150-191.
[22] J.L. Colliot-Thelene and W. Raskind, K2-cohomology and the second Chow group, Math.
Ann. 270 (1985) 283-318.
[23] J.-L. Colliot-Thelene and J.-J. Sansuc, Torseurs sous des groupes de type multiplicatif, C.R.
Acad. Sci. Paris 282A (1976) 1113-1116.
[24] J.-L. Colliot-Thelene and J.-J; Sansuc, Varietes de premiere descente attachees aux varietes
rationnelles, C.R. Acad. Sci. Paris 284A (1977) 967-970.
[25] J.-L. Colliot-Thelene and J.-J. Sansuc, La descente sur une variete rationnalle definie sur un
corps de nombres, C.R. Acad. Sci. Paris 284A (1977) 1215-1218.
[26] J.-L. Colliot-Thelene and J.-J. Sansuc, La R-equivalence sur les tores, Ann. Sci. Ec. Norm.
Sup. Ser. 4 10 (1977) 175-229.
[27] J.-L. Colliot-Thelene and J.-J. Sansuc, La descente sur les varietes rationnelles, in: Journee
de Geometric Algebrique d’Angers, 1979 (Sijthoff-Noordhoff, Alphen aan den Rijn, 1980)
223-237.
[28] J.-L. Colliot-Thelene and J.-J. Sansuc, On the Chow groups of certain rational surfaces: a
sequel to a paper of S. Bloch, Duke Math. J. 48 (1981) 421-447.
[29] J.-L. Colliot-Thelene and J.-J. Sansuc, Sur le principe de Hasse et l’approximation faible, et
sur une hypothese de Schinzel, Acta Arithm. 41 (1981) 33-53.
[30] J.-L. Colliot-Thelene, J.-J. Sansuc and Ch. Soule, Quelques theoremes de finitude en theorie
des cycles algebriques, C.R. Acad. Sci. Paris 294 (1982) 749-752.
[31] J.-L. Colliot-Thelene, J.-J. Sansuc and Ch. Soule, Torsion dans le groupe de Chow de
codimension deux, Duke Math. J. 50 (1983) 763-801.
[32] J.-L. Colliot-Thelene, J.-J. Sansuc and P. Swinnerton-Dyer, Intersections de deux quadriques
et surfaces de Chatelet, C.R. Acad. Sci. Paris 298 (1984) 377-380.
[33] K.R. Coombes and D. J. Muder, Zero-cycles on Del Pezzo surfaces over local fields, Preprint,
1983.
[34] D. Coray, Arithmetic on singular cubic surfaces, Compos. Math. 33 (1976) 55-67.
[35] D. Coray, Points algebriques sur les surfaces de Del Pezzo, C.R. Acad. Sci. Paris 284 (1977)
1531-1534.
[36] D. Coray, The Hasse principle for pairs of quadratic forms, in: Journee Arithm. (Cambridge j
Univ. Press, 1982)238-245.
[37] D. Coray and M. Tsfasman, Arithmetic on singular Del Pezzo surfaces, Preprint, 1984.
[38] M. Demazure, Sous-groups algebriques de rang maximum du group de Cremona, Ann. Sci.
ENS. Ser. 4 3(1970) 507-588.
[39] M. Demazure, Surfaces de Del Pezzo II, III, IV, V., Lecture Notes in Math. 777 (Springer,
Berlin, 1980) 23-69.
[40] P. Du Vai, On isolated singularities which do not affect the conditions of adjunction I, II, III,
Proc. Cambridge Phil. Soc. (30) 1934,453-465, 483-491.
[41] El Huti, Cubic surfaces of Markov type, Mat. Sbornik 93(3) (1974) 331-346 (in Russian).
[42] S. Endo and T. Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973)
7-26.
[43] S. Endo and T. Miyata, Quasi-permutation modules over finite groups I, II, J. Math. Soc.
Japan 25 (1973) 397-421; 26 (1974) 698-713.
[44] S. Endo and T. Miyata, On a classification of the function fields of algebraic tori, Nagoya
Math, J. 56 (1975) 85-104.
[45] G. Fano, Sopra alcune varieta algebriche a tre dimensioni aventi tutti i generi nulli, Atti Acc.
Torino 43 (1908)973-977.
[46] G. Fano, Osservazioni sopra alcune varieta non razionali aventi tutti i generi nulli, Atti Acc.
Torino 50 (1915) 1067-1072.
[47] G. Fano, Nuove ricerche sulle varieta algebriche a tre dimensioni a curve-serioni canoniche,
Comm. Pont. Acad. Sci. (1947) 635-720.
[48] M.H. Gizatullin, Rational G-surfaces. Izv. AN SSSR, Ser. Mat. 41(1) (1980) 110-144 (in
Russian).
[49] M.H. Gizatullin, Defining relations for the plane Cremona group, Izv. AN SSSR, Ser. Mat.
46(5) (1982) 909-970.
[50] D.R. Heath-Brown, Cubic forms in ten variables, Proc. London Math. Soc. 47(2) (1983)
225-257.
[51] J.W.P. Hirschfeld, Del Pezzo surfaces over finite fields, Tensor 37 (1982) 79-84.
[52] W. Hiirlimann, Brauer group and diophantine geometry: a cohomological approach, Leet.
Notes Math. 917 (Springer, Berlin, 1981).
[53] V. A. Iskovskih, Rational surfaces with pencil of rational curves, Mat. Sb. 74(4) (1967)
608-638 (in Russian).
[54] V. A. Iskovskih, Rational surfaces with pencil of rational curves and positive square of
canonical class, Mat. Sb. 83(1) (1970) 90-119 (in Russian).
[55] V.A. Iskovskih, A counter-example for the Hasse principle for a system of two quadratic
forms in five variables, Mat. Zametki 10 (1971) 253-257 (in Russian).
[56] V.A. Iskovskih, Birational properties of surfaces of degree four in P4, Mat. Sb. 88(1) (1972)
31-37 (in Russian).
[57] V.A. Iskovskih, Three-dimensional Fano varieties. I, Izv. AN SSSR, Ser. Mat. 41(3) (1977)
516-562 (in Russian).
[58] V.A. Iskovskih, Three-dimensional Fano varieties. II, Izv. AN SSSR, Ser. Mat. 42(3) (1978)
504-549 (in Russian).
[59] V.A. Iskovskih, Minimal models of rational surfaces over arbitrary fields, Izv. AN SSSR, Ser.
Mat. 43 (1979) 19-43 (in Russian).
[60] V.A. Iskovskih, Anticanonical models of three-dimensional algebraic varieties, in: Sovremen-
nye Problemy Matematiki 12 (VINITI, Moscow, 1979) 59-157 (in Russian).
[61] V.A. Iskovskih, Birational automorphisms of three-dimensional algebraic varieties, in:
Sovremennye Problemy Matematiki 12 (VINITI, Moscow, 1979) 159-236 (in Russian).
[62] V.A. Iskovskih, Birational automorphisms of three-dimensional algebraic varieties close to
rational ones, in: Algebra, Vol. 1, Moscow University (1980) 40-49 (in Russian).
[63] VA. Iskovskih, Generators and relations in the two-dimensional Cremona group, Vestnik
MGU, Ser. 1 5 (1983) 43-48 (in Russian).
[64] V.A. Iskovskih, Generators and relations in the groups of birational automorphisms of two
classes of rational surfaces, Trudy Steklov Inst. 165 (1984) 67-78 (in Russian).
[65] V A. Iskovskih and Yu.I. Manin, Three-dimensional quartics and counter-examples to the
Liiroth problem, Mat. Sb. 86 (1971) 140-166 (in Russian).
[66] D.S. Kanevski, Classes of points on cubic hypersurfaces, defined over finite fields, Mat.
Zametki 20(1) (1976) 121-131 (in Russian).
[67] D.S. Kanevski, Structure of groups, related to cubic surfaces, Mat. Sb. 103(2) (1977) 292-308
(in Russian).
[68] D. Kanevsky, On cubic planes and groups connected with cubic surfaces, J. Algebra 80(2)
(1983)559-565.
[69] D. Kanevsky, Some remarks on Brauer equivalence for cubic surfaces, Preprint, Max Planck
Inst, fur Math., Bonn, 1983.
[70] B.E. Kunyavski, On tori with biquadratic splitting fields, Izv. AN SSSR, Ser. Mat. 42(3)
(1978) 537-542 (in Russian).
[71] B.E. Kunyavski and M.A. Tsfasman, Zero-cycles on rational surfaces and Neron-Severi tori,
Izv. AN SSSR, Ser. Mat. 48 (1984) 631-654 (in Russian).
[72] H.W. Lenstra, Rational functions invariant under a finite abelian group, Invent. Math. 25
(1974)299-325.
[73] E. Looijenga, On the semiuniversal deformation of a simple elliptic hypersurface singularity.
II, Topology 17 (1978) 23-40.
[74] J.-Y. Merindol, Les singularites simples elliptiques, leurs deformations, les surfaces de Del
Pezzo et les transformations quadratiques, Ann. Sci. ENS15 (1982) 17-44.
[75] A.S. Merkur’ev and A.A. Suslin, X-cohomology of Severi-Brauer varieties and norm
residue homomorphism, Izv. AN SSSR, Ser. Mat. 46(5) (1982) 1011-1046 (in Russian).
[76] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math.
116(1982)133-176.
[77] Sh. Mukai and H. Umemura, Minimal rational threefolds, in: Algebraic Geometry, Lecture
Notes in Math. 1016 (Springer, Berlin, 1983) 490-517.
[78] J.P. Murre, Reduction of the proof of the nonrationality of a nonsingular cubic threefold to a
result of Mumford. Compos. Math. 27 (1973) 63-82.
[79] I. Naruki and T. Urabe, On singularities on degenerate Del Pezzo surfaces of degree 1,2,
Preprint, 1981.
[80] H. Pinkham, Simple elliptic singularities, Del Pezzo surfaces and Cremona transformations,
Proc. Symp. Pure Math. 30 (1977) 65-84.
[81] V.P. Platonov and A.S. Rapinchuk, Algebraic groups, in: Algebra. Topology. Geometry,
Vol. 21 (VINITI, Moscow, 1983) 80-134 (in Russian).
[82] V.P. Platonov, Arithmetic theory of algebraic groups, Usp. Mat. Nauk 37(3) (1982) 3-54 (in
Russian).
[83] J.J. Sansuc, Descente et principe de Hasse pour certaines varietes rationnelles, Sem.
Delange-Pisot-Poiton (1980-81) 254-271.
[84] J.-J. Sansuc, Groupe de Brauer et arithmetique de groupes algebriques lineaires, J. Reine
Angew. Math. 327 (1981) 12-80.
[85] J.-J. Sansuc, A propos d’une conjecture arithmetique sur le groupe de Chow d’une surface
rationnelle, in: Seminaire de Theorie desNombres de Bordeaux, Vol. 33 (1981-1982).
[86] V. Sarkisov, Birational automorphisms of conic bundles, Izv. AN SSSR 17 (1981) 177-202 (in
Russian).
[87] V.V. Sokurov, Existence of a line on the Fano varieties, Izv. AN SSSR, Ser. Mat. 43(4) (1979)
922-964 (in Russian).
[88] V.V. Sokurov, Smoothness of a generic anticanonical divisor or the Fano varieties, Izv. AN
SSSR, Ser. Mat. 43 (1979) 430-441 (in Russian).
[89] J.D.H. Smith, On the nilpotence class of commutative Moufang loops, Math. Proc.
Cambridge Phil. Soc. 84 (1978) 387-404.
[90] J.D.H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67 (1982)
173-187.
[91] R.G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969)
148-158.
[92] H.P.F. Swinnerton-Dyer, Two special cubic surfaces, Mathematika 9 (1962) 54-56.
[93] H.P.F. Swinnerton-Dyer, Rational points on Del Pezzo surfaces of degree 5, in: Proc. 5th
Nordic Summer School in Math. Oslo (1970) 287-290.
[94] H.P.F. Swinnerton-Dyer, The birationality of cubic surfaces over a given field, Michigan
Math. J. 17(1970)289-295.
[95] H.P.F. Swinnerton-Dyer, Universal equivalence for cubic surfaces over finite and local fields,
Symp. Math., Bologna 24 (1981) 111-143.
[96] G. Timms, The nodal cubic surfaces and the surfaces from which they are derived by
projection, Proc. Royal Soc. London A119 (1928) 213-248.
[97] M.A. Tsfasman, Arithmetic of conic bundles, Usp. Mat. Nauk 37(2) (1982) 239-240 (in
Russian).
[98] A.N. Turin, Five lectures of three-dimensional varieties, Usp. Mat. Nauk 27(5) (1972) 3-50
(in Russian).
[99] A.N. Turin, Intermediate Jacobian of three-dimensional varieties, in: Sovremennye Prob-
lemy Matematiki, Vol. 12 (VINITI, Moscow, 1979) 5-57.
[100] VE. Vbskresenski, On the birational equivalence of linear algebraic groups, Doklady AN
SSSR 188(5) (1969) 978-981 (in Russian).
[101] VE. Voskresenski, Birational properties of linear algebraic groups, Izv. AN SSSR, Ser. Mat.
34(1) (1970) (in Russian).
[102] V.E. Voskresenski, Rationality of certain algebraic tori, Izv. AN SSSR, Ser. Mat. 35(5)
(1971) (in Russian).
[103] V.E. Voskresenski, Stable equivalence of algeraic tori, Izv. AN SSSR, Ser. Mat. 38 (1974)
3-10 (in Russian).
[104] V.E. Voskresenski, On birational invariants of algebraic tori, Usp. Mat. Nauk 30(2) (1975)
207-208 (in Russian).
[105] V.E. Voskresenski, Algebraic Tori (Moscow, Nauka, 1977) (in Russian).
[106] V.E. Voskresenski, Problems of rationality of linear algebraic groups, Usp. Mat. Nauk 32(2)
(1977) 241 (in Russian).
[107] V.E. Voskresenski, On the reduced Whitehead group of a simple algebra, Usp. Mat. Nauk
32(6) (1977) 247-248 (in Russian).
[108] V.E. Voskresenski, On the R-equivalence of semisimple groups. Zapiski Nauc. Sem. LOMI
86 (1979) 49-65 (in Russian).
[109] V.E. Voskresenski, Projective invariant Demezure’s models, Izv. AN SSSR, Ser. Mat. 46(2)
(1982) 195-210 (in Russian).
[110] V.E. Voskresenski and A. A. Kliachko, Toric Fano varieties and root systems, Izv. AN SSSR
48(2) (1984) 237-263 (in Russian).
REFERENCES
Algebraic Number Theory, eds. J.W.S. Cassels and A, Frohlich (Academic Press,
New York, 1967).
Algebraic Surfaces, Tr. Mat. Inst. Steklov 75 (1965) (English TransL: Proc. Steklov
Inst. Math. 75, Am. Math. Soc., Providence, R.I., 1968).
E. ARTIN
[1] Geometric algebra (Interscience, New York, 1957).
M. ARTIN
[1] Some numerical.criteria for contractibility of curves on algebraic surfaces, Am. J.
Math. 84(1962)485-496.
S. BECKEN
[1] Spiegelungsrelationen in orthogonalen Gruppen, J. Reine Angew. Math. 210 (1962)
205-215.
V.D. BELOUSOV
[1] On the structure of distributive quasigroups, Mat. Sborn. 50 (3) (1960) 266-298
(in Russian).
E. BOMBIERI and H.P.F. SWINNERTON-DYER
[1 ] On the local zeta function of a cubic threefold, Ann. Scuola Norm. Sup. Pisa (3) 21
(1967) 1-29.
N. BOURBAKI
[1] Groupes et algebres de Lie (Hermann, Paris, 1968) chs. 4 —6.
[2] Algebre (Hermann, Paris, 1958, 1959) chs. 1-9.
R. BRAUER and M. SUZUKI
[ 1 ] On finite groups whose 2-Sylow group is a quaternion group, Proc. Natl. Acad. Sci.
USA 45 (1959) 1757-1759.
R.H. BRUCK
[1] Some results in the theory of quasigroups, Trans. Am. Math. Soc. 55 (1944) 19—52.
[2] A survey of binary systems (Springer, Berlin, 1958).
[3] An open question concerning Moufang loops, Arch. Math. (Basel) 10 (1959) 419-421.
H. CARTAN and §. EILENBERG
[1] Homological algebra (Princeton Univ. Press, Princeton, N.J., 1956).
P. CARTIER
[1J Diviseurs amples, Sem. Bourbaki (1966) expose 301.
J.W.S. CASSELS
[1] Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung, J. Reine Angew.
Math. 211 (1962)95-112.
[2] Arithmetic on curves of genus 1. VI. The Tate—SafareviC group can be arbitrarily large,
J. Reine Angew. Math. 214/215 (1964) 65-70.
[3] Diophantine equations with special reference to elliptic curves, J. London Math. Soc.
41 (1966) 193-291. (Corrigenda: ibid. 42 (1967) 183.)
J.W.S. CASSELS and M.J.T. GUY
[1] On the Hasse principle for cubic surfaces, Matematika 13 (1966) 111—120.
N.T. Cebotarev
[1] Introduction to the theory of algebras (Ogiz, Moscow, 1949) (in Russian).
F. CHATELET
[1] Points rationnels sur certaines courbes et surfaces cubiques, Enseignement Math. 5
(1959) 153-170.
[2] Points rationnels sur certaines surfaces cubiques, Colloq. CNRS Clermont-Ferrand,
1964 (Paris, 1966) 67-75.
C. CHEVALLEY
[1] Sem. C. Chevalley, Anneaux de Chow et applications (Secretar. Math., Paris, 1958).
P. DELIGNE
[ 1 ] Intersections sur les surfaces regulieres, SGA 7 (Inst. Hautes Etudes Sci., 1969) expose
10.
A. DICKSON
[1] History of the theory of numbers, Vol. 2. Diophantine analysis (Washington, 1920).
G. FANO
[1] Su alcune varieta algebraiche a tre dimension! aventi curve-sezioni canoniche, Scritti
Mat. offerti a L. Bersolari (Pavia, 1936) 329—349.
[2] Sulle varieta algebraiche a tre dimension! a curve-sezioni canoniche, Mem. Accad.
Italia 8 (1937) 23-64.
[3] Su alcune varieta algebraiche a tre dimension! razionali, e aventi curve-sezioni
canoniche, Comment. Math. Helv. 14 (1941) 202-211.
[4] Nuove ricerche sulle varieta algebraiche a tre dimension! a curve-sezioni canoniche,
Comment. Pont. Acad. Sci. 11 (1947) 635—720.
B. FISCHER
[1] Distributive Quasigruppen endlicher Ordnung, Math. Z. 83 (1964) 267-303.
J.S. FRAME
[ 1 j The classes and representations of the groups of 27 lines and 28 bitangents, Ann.
Mat. Рига Appl. (4) 32 (1951) 83-119.
F. GHER ALDELLI
[1] Un osservazione sulla varieta cubica di P3, Rend. Sem. Mat. Fis. Milano 37 (1967)
157-160.
M. GREENBERG
[1] Lectures on forms in many variables (Benjamin, New York, 1970).
A. GROTHENDIECK
[1] Sur quelques proprietes fondamentales en theorie des intersections, Sem. C. Cheval-
ley, Anneaux de Chow et applications (Secretar. Math., Paris, 1958).
[2] Le groupe de Brauer, I, II, III, in: Dix exposes sur la cohomologie des schemas
(North-Holland, Amsterdam, 1968) 46—66, 67-87, 88-188.
A. GROTHENDIECK and J. DIEUDONNE
[1] Elements de geometric algebrique, Publ. Math. I.H.E.S. 4, 8,. .. (1960,... ).
M. HALL
[1] Theory of groups (MacMillan, New York, 1959).
[2] Automorphisms of Steiner triple systems, Proc. Symp. Pure Math., Vol. 6 (Am.
Math. Soc., Providence, R.I., 1962) 47 -66.
R. HARTSHORNE
[1] Curves with high self-intersection on algebraic surfaces, Publ. Math. I.H.E.S.
36(1969) 111-125.
A. HENDERSON
[1] The twenty-seven lines upon the cubic surface (Cambridge Univ. Press, Cambridge,
1911).
C. JORDAN
[1] Traite des substitutions et des equations algebriques (Gauthier-Villars, Paris, 1957).
S. KLEIMAN
[1 ] Algebraic cyclesand the Weil conjectures, in: Dix exposes sur la cohomologie des
schemas (North-Holland, Amsterdam, 1968) 359-386.
S. LANG
[1] Diophantine geometry (Interscience, New York, 1962).
J. LIPMAN
[1J Rational singularities with applications to algebraic surfaces and unique factorization,
Publ. Math. LH.E.S. 36 (1969) 195-279.
Yu. I. MANIN
[1] On the arithmetic of rational surfaces, Dokl. Akad. Nauk SSSR 152 (1963) 46 -49
(in Russian).
[2] The Tate height of points on an Abelian variety, its variants and applications, Izv.
Akad. Nauk SSSR Ser. Mat. 28 (1964) 1363—1390 (in Russian).
[3] Rational surfaces over perfect fields, Publ. Math. LH.E.S. 30 (1966) 55 -113 (in
Russian).
[4] Rational surfaces over perfect fields, Mat. Sb. 72 (114) (1967) 161-192 (in Russian).
[5] Two theorems on rational surfaces, in: Proc. Symp. Int. Geometria Algebraica
(Rome, 1967) 198-207.
[6] Cubic hypersurfaces I. Quasigroups of classes of points, Izv. Akad. Nauk SSSR Ser.
Mat. 32 (1968) 1223 -1244 (in Russian).
[7] Correspondences, motives and monoidal transformations, Mat. Sb. 77 (1968)
475—507 (in Russian).
[8] Hypersurfaces cubiques IL Automorphismes birationnels en dimension deux, Invent,
Math. 6 (1969)334-352.
[9] Cubic hypersurfaces III. Moufang loops and Brauer equivalence, Mat. Sb. 79 (121)
(1969) 155 — 170 (in Russian).
W.F. MEYER
[1] Flachen dritter Ordnung, Encyclopadie Math. Wiss. Bd. HI, 2 h.ll, IIIc 10a (Berlin,
1928).
L.J. MORDELL
[1] On the conjecture for the rational points on a cubic surface, J. London Math. Soc.
40(1965) 149 — 158.
[2] Diophantine equations (Academic Press, New York, 1969).
D. MUMFORD
[1] A remark oh Mordell’s conjecture, Am. J. Math. 87 (1965) 1007 —1016.
12] Lectures on Curves on an algebraic surface (Princeton Univ. Press, Princeton, N.J.,
1966).
M. NAGATA
[1] On rational surfaces, I, II, Mem. Coll. Sci. Univ. Kyoto, Ser. A, Math. 32 (1960)
351 -370; ibid. 33 (1960/61) 271 -293.
T. ONO
[ 1) Arithmetic of algebraic tori, Ann. Math. 74 (1961) 101 -139.
[2J. On the Tamagawa number of algebraic tori, Ann. Math. 78 (1963) 47—73.
P. ROQUETTE
[1] On the Galois cohomology of the projective linear group and its applications to the
construction of generic splitting fields of algebras, Ann. Math, 150 (1963)411—439.
L. ROTH
[1] Algebraic threefolds (with special regards to problems of rationality) (Springer, Berlin,
1955).
LR. SAFAREVlt
[1] Lectures on minimal models and birational transformations of two-dimensional
schemes (Tata Inst. Fund. Res., Bombay, 1966).
[2] Fundamentals of algebraic geometry, Usp. Mat. Nauk 24 (1969) 3—184 (in Russian).
S. SCHANUEL
[1] Heights in number fields, Bull. Am. Math. Soc. 70 (1964).
B. SEGRE
[1J The non-singular cubic surfaces (Clarendon, Oxford, 1942).
[2] A parametric solution of the indeterminate cubic equation z2 -/(x, 7), J. London
Math. Soc. 18 (1943) 226 - 233.
[3] The rational solutions of homogeneous cubic equations in four variables, Math.
Notae Univ. Rosario anno II, fasc. 1—2 (1951) 1—68.
[4] Arithmetical questions on algebraic varieties (London, 1951).
E. SELMER
[1] Sufficient congruences for the existence of rational points on certain cubic surfaces,
Math. Scand. 1 (1953) 113-119.
A.M. SERMENEV
[1] On some unirational surfaces, Mat. Zametki 5 (1969) 155—159 (in Russian).
[2] On motives of cubic hypersurfaces, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970)
515—522 (in Russian).
J.P. SERRE
[1] Critere de rationalite pour les surfaces algebriques, Sem. Bourbaki (1957).
[2] Morphismes universels et difference Iles de troisieme espece, Sem. C.Chevalley,
Anneaux de Chow et applications (Secretar. Math., Paris, 195 8).
[3] Cohomologie Galoisienne (Springer, Berlin, 1964).
[4] Lie algebras and Lie groups (Benjamin, New York, 1965).
[5] Zeta functions and L-functions, in: Arithmetical algebraic geometry (Harper and
Row, London, 1965) 82—92.
[6] Local class field theory, in: J.W.S. Cassels and A. Frohlich, eds., Algebraic number
theory (Academic Press, New York, 1967).
[7] Abelian Z-adic representations and elliptic curves (Benjamin, New York, 1968).
SNYDER
[1] The problem of the cubic variety in P3, Bull. Am. Math. Soc. 35 (1929) 607 —642.
HJP.F. SWINNERTON-DYER
[1] Two special cubic surfaces, Matematika 9 (1962) 54—56.
[2] The zeta function of a cubic surface over a finite field, Proc. Cambridge Phil. Soc.
63(1967) 55-71.
J.TATE
[1 ] On the conjectures of Birch and Swinnerton-Dyer and a geometric analogue, in:
Dix exposes sur la cohomologie des schemas (North-Holland, Amsterdam, 1968)
189-214.
J.A. TODD
[1] On the topology of certain threefold loci, Proc. Edinburg Math. Soc. Ser. 2,4 (1935)
175-184.
A. TURIN
[1] On non-singular Fano cubic surfaces in P3, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970)
1200—1208 (in Russian).
V. VOSKRESENSKII
[1] On two-dimensional algebraic tori, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965)
230—244 (in Russian).
R. WALKER
(1] Algebraic curves (Dover Publ., New York, 1962).
A. WEIL
[1 ] Arithmetic on algebraic varieties, Ann. Math. 53 (1951) 412-444.
[2] Adbles and algebraic groups, Matematika 8 (1964) 3—74 (a translation into Russian
of mimeographed notes).
[3] Abstract versus classical algebraic geometry, Proc. Int. Congr. of Mathematicians,
Amsterdam, Vol. 3 (1964) 550-558.
O. ZARISKI
[ 1 ] Introduction to the problem of minimal models in the theory of algebraic surfaces
(Tokyo, 1958).
AUTHORINDEX
Artin, E., 181,187
Artin, M., 105
Becken, S., 187
Belousov, V.D., 10, 39
Bombieri, E., 59
Bourbaki, N„ 116,128,129,133,139,
220
Brauer, R., 39
Bruck, R.H., 8,10,39,40
Cartan, H„ 180
Cartier, P., 125, 152
Cassels, J.W.S., 3,167,189, 224, 254,
269, 280, 283
Cebotarev, N.T., 220, 230
Chatelet, F., 5, 255
Deligne, P., 189
Dickson, A., 1
Dieudonne, J., see A. Grothendieck
Eilenberg, S., see H. Car tan
Fano, G., 189
Fischer, B., 10,19,39
Frame, J.S., 152
Greenberg, M., 174
Grothendieck, A., 5, 147, 222, 229,
283
Guy, M.J.T., see J.W.S. Cassels
Hall, M., 10,21,35,39,40
Hartshorne, R., 81,111
Henderson, A., 112,182,183
Jordan, C., 182
Kleiman, S., 182
Lang, S., 267
Lipman, J., 111,182
Manin, Yu.L, 39,57,76,111,138,
147, 179, 182, 183, 203, 219, 267,
283
Meyer, W.F., 183 '
Mordell, L.J., 224, 276
Mumford, D., 5,90,91,96,97,270
Nagata, M., Ill
Ono, T., 166,171
Roquette, P., 159
Roth, L., 189
Safarevii!, I.R., 5, 102,104,105, 110,
111,117,183,247
Schanuel, S., 270
Segre, B., 5,110,112,114,117,182,
219
Sermenev, A.M., 162
Serre, J.P., 112,116,117,127, 132,
133, 139,175, 220, 240
Suzuki, M., see R. Brauer
Swinnerton-Dyer, H.P.F., 152,174,
175, 182, 224, 278
Swinnerton-Dyer, H.P.F., see E. Bom-
bieri
Tate, J., 181
Todd, J.A., 117
Voskresenskil, V., 172,166
Walker, R., 15
Weil, A., 112,182,242
Zariski, O., 111
LIST OF SYMBOLS
fi(x) — p. 16
C(x) — p. 51
f*(P) -p. 91
f~HD) — p. 92
4 ~p. 114
MD -p. 112
7Vr -p. 114
R(-equivalence) -p. 61
RK/k -p. 67
Rr -p. 114
sx,y -p. 185
tx (reflection in a symmetric quasigroup) -p. 11
tx (birational map) — pp. 14, 53
T(K) -p. 11
7°(£) -p. 11
x о у (composition in a symmetric quasigroup) -p. 6
x о у (composition of points) — p. 55
X о Y (composition of classes of points) ~ pp. 45, 55
SUBJECT INDEX
Abelian (symmetric) quasigroup 7, 1 Iff
Admissible equivalence relation 43ff
Associative centre 9
Associator 23
Azumaya algebra 221, 229ff
^-equivalence 223, 227, 234ff
Birational map 47, lOOff
Birational triviality 48
Blowing up of a point 7 8
Brauer equivalence 44, 220ff, 234ff
Brauer-Grothendieck group 220ff
Brauer group 220ff
Bubble space 195ff
Cartier divisor 90
Cassels—Tate form 227ff
Centre of a CML 9
CH-quasigroup 8, 15ff
Chatelet surface 255
CML 8, 21ff
Collapsing of a curve 78
Collinearity 6, 7, 42
Commutative Moufang loop 8, 2Iff
Conical singular point 48
Del Pezzo surface 117ff
Distributive (symmetric) quasigroup 10,
30
Divisor 94
Dominating map 4 7
Effective (Cartier) divisor 90
Effective (Weil) divisor 91
Exceptional class 134
Exceptional curve 78,106
Exceptional subset 140
Fischer group 20,34ff
Fisher’s theorems 34, 37
Fundamental birational invariant 189ff
General position, points in 54
General type, point of 51
Good point (pair of points) 185
Hasse principle 165, 166, 171, 172, 224,
276ff
Height 267ff
Index of a morphism 77
Index of a surface 165
Intersection number 90, 96
k- topology 70
Minimal model 80
Minimal surface 79, 107, 184ff, 218
Model of a field 79
Monoidal transformation 78, 82ff, 85
Moufang’s theorem 22
Multiplicity of a point on a divisor 92
Picard group 77,95,126
Point in general position 54
Point of general type 51
Point over a surface 196
Principal homogeneous space 167
Proper inverse image of a divisor 92
Rational map 46
Reflection 133, 187
R-equivalence 44, 61 ff
Resolution of singularities of a map 82ff,
lOOff
§afarevi&—Tate group 228
Schlaffli sixfold 173
326 SUBJECT INDEX
^-exceptional set 152 Seven-Brauer surface 159,169 Special variety 61 Symmetric quasigroup 6 Universal equivalence relation 43, 44, 54ff, 69ff Weil divisor 91 Weil height 268
Torus 166 Weyl group 133ff, 151ff
Unirationality 46ff, 50, 154ff Zeta function 143