Arithmetic and Geometry: Papers Dedicated to I.R. Shafarevich on the Occasion of His Sixtieth Birthday Volume I Arithmetic

184 Pages·1983·10.68 MB·English

by  Noboru Aoki

Progress in Mathematics
Vol. 35
Edited by
J. Coates and
S. Helgason
Arithmetic and
Geometry
Papers Dedicated to
I.R. Shafarevich on the Occasion
of His Sixtieth Birthday
Volume I Arithmetic
Michael Artin,
John Tate,
editors
Birkhauser
Boston - Basel. Stuttgart Birkhauser
Boston Basel Stuttgart
Editors:
Michael Artin John Tate
Mathematics Department Mathematics Deparhnent
Massachusetts Institute of Technology Harvard University
Cambridge, MA 02139 Cambridge, MA 02138
Igor Rostislavovich Shafarevich has made outstanding contribu-
This book was typeset at Stanford University using the TEX document preparation tions in number theory, algebra, and algebraic geometry. The flour-
system and computer modern type fonts by Y. Kitajima. Special thanks go to ishing of these fields in Moscow since World War I1 owes much to his
Donald E. Knuth for the use of this system and for his personal attention in the de- influence. We hope these papers, collected for his sixtieth birthday,
velopment of additional fonts required for these volumes. In addition, we extend
will indicate to him the great respect and admiration which mathema-
thanks to the contributors and editors for their patience and gracious help with im-
ticians throughout the world have for him.
plementing this system. Michael Artin
Igor Dolgachev
Library of Congress Cataloging in Publication Data John Tate
Main entry under title: A.N. Todorov
Arithmetic and geometry.
(Progress in mathematics ; v. 35-36)
Contents: v. 1. Arithmetic - v. 2. Geometry.
1. Algebra- Addresses, essays, lectures. 2. Geo-
metry, Algebraic- Addresses, essays, lectures.
3. Geometry - Addresses, essays, lectures. 4. Shafare-
vich, I. R. (Igor' Rostislavovich), 1923-
I. Shafarevich, I. R. (Igor' Rostislavovich), 1923-
11. Artin, Michael. 111. Tate, John Torrence,
1925- . IV. Series: Progress in mathematics
(Cambridge, Mass.) ; v. 35-36.
QA7.A67 1983 513'.132 83-7124
ISBN 3-7643-3132-1 (v. 1)
ISBN 3-7643-3133-X (v. 2)
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Arithmetic and geometry : papers dedicated to
I. R. Shafarevich on the occasion of his 60.
birthday 1 Michael Artin ; John Tate, ed. -
Boston ; Basel ; Stuttgart : Birkhauser
(Progress in mathematics ; ...)
NE: Artin, Michael (Hrsg.); Safarevic, Igor' R.:
Festschrift
Vol. 2. Geometry h
All rights reserved. No part of this publication may be reproduced, stored in a re-
trieval system, or transmitted, in any form or by any means, electronic, mechani-
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Birkhauser Boston, Inc., 1983
@
ISBN 3-7643-3 132-1
Printed in USA
I.R. Shafarevich, Moscow, 1966
Volume I Arithmetic
Volume I1 Geometry
N. Aoki and T. Shioda, Generators of the NCron-Severi Group of a
V.I. Arnold, Some Algebro-Geometrical Aspects of the Newton
Fermat Surface
Attraction Theory
S. Bloch, p-adic Etale Cohomology
M. Artin and J. Denef, Smoothing of a Ring Homomorphism Along a
J.W.S. Cassels, The Mordell-Weil Group of Curves of Genus 2 Section
G.V. Chudnovsky, Number Theoretic Applications of Polynomials with M.F. Atiyah and A.N. Pressley, Convexity and Loop Groups
Rational Coefficients Defined by Extremality Conditions
H. Bass, The Jacobian Conjecture and Inverse Degrees
J. Coates, Infinite Descent on Elliptic Curves with Complex
R. Bryant and P. Griffiths, Some Observations on the Infinitesimal
Multiplication
Period Relations for Regular Threefolds with Trivial Canonical Bundle
N.M. Katz, On the Ubiquity of Pathology in Products
H. Hironaka, On Nash Blowing-Up
S. Lang, Conjectured Diophantine Estimates on Elliptic Curves
F. Hirzeb~ch,A rrangements of Lines and Algebraic Surfaces
S. Lichtenbaum, Zeta-Functions of Varieties over Finite Fields at s = 1 V.G. Kac and D.H. Peterson, Regular Functions on Certairi Infinite-
B. Mazur and J. Tate, Canonical Height Pairings via Biextensions dimensional Groups
J.S. Milne, The Action of an Automorphism of C on a Shimura Variety W.E. Lang, Examples of Surfaces of General Type with Vector Fields
and its Special Points
Yu.1. Manin, Flag Superspaces and Supersymmetric Yang-Mills
N.O. Nygaard, The Torelli Theorem for Ordinary K3 Surfaces over Equations
Finite Fields
B. Moishezon, Algebraic Surfaces and the Arithmetic of Braids, I
A.P. Ogg, Real Points on Shimura Curves
D. Mumford, Towards an Enumerative Geometry of the Moduli Space
1.1. Piatetski-Shapiro, Special Automorphic Forms on PGSp, of Curves
M. Raynaud, Courbes sur une variCtC abClienne et points de torsion C. Musili and C.S. Seshadri, Schubert Varieties and the Variety of
Complexes
A. Weil, Euler and the Jacobians of Elliptic Curves
A. Ogus, A Crystalline Torelli Theorem for Supersingular K3 Surfaces
M. Reid, Decomposition of Toric Morphisms
M. Spivakovsky, A Solution to Hironaka's Polyhedra Game
A.N. Tjurin, On the Superpositions of Mathematical Instantons
A.N. Todorov, How Many Kahler Metrics Has a K3 Surface?
0. Zariski, On the Problem of Irreducibility of the Algebraic System
of Irreducible Plane Curves of a Given Order and Having a Given
Number of Nodes
Generators of the NCron-Severi Group
of a Fermat Surface
Noboru Aoki and Tetsuji Shioda
To Z.R. Shafarevich
1. Introduction
The Nkron-Sevcri group of a (nonsingular projective) varicty is, by
definition, the group of divisors rnodulo algebraic equivalence, which is
known to be a fiuitcly generated abelian group (cf. [2]). Its rank is called
the Picard number of the variety. Thus the Nkron-Severi group is defined
in purely algebro-geometric terms, but it is a rather delicate invariant of
arithmetic nature. Perhaps, because of this reason, it usually requires some
nontrivial work before one can determine the l'icard number of a given
variety, let alone the full structure of its N6ron-Severi group. This is the
case even for algebraic surfaces over the field of complex numbers, where
it can be regarded as the subgroup of the cohornology group I12(X,Z )
characterized by the LefscheLz criterion.
NOWt he purpose of the present paper is to find certain explicitly defined
curves on the complex Fermat surface
whose cohomology classes (equivalently, algebraic equivalence classes) form
(a part of') gcrlcrators of the Ntkon-Severi group NS (X&)@Q . The Pirard
number p(Xk)h as recently been determined and is given by the following
for~nula:
where
AOKI AND SHIODA GENERATORS OF THE N~RON-SEVERIG ROUP 3
{
1 (m : even) Futhermore, it is expected that a similar approach should be applicable
L =
to Fermat varieties of higher dimension for the explicit construction of
0 (m : odd)
{ 1 algebraic cycles corresponding to given Hodge classes. Combined with the
:I3 (if 3 m)
(m/3)* = method based on the inductive structure of Fermat varieties (31, this might
(if 3 m)
{ lead to the verification of the Hodge Conjecture for all Fermat varieties.
:I2 (m : even)
(m/2)* =
(m : odd)
and where c(m) is a bounded function of m which is expressed as a certain 2. Preliminaries
sum over divisors d of m such that (dl 6) # 1 and d 5 180 (see Shioda
+ +
[4], Aoki[l]). It is known that 3(m - l)(m - 2) 1 6, is the rank of the First recall that the Hodge classes on the Fermat variety
subspace of NS (XL) which is spanned by the lines of the ambient space Xk : zy = 0 are described in terms of the characters of the group
P3 lying on XL; in particular, if (m, 6) = 1, then NS (X&)@Q is spanned Gk = (I.~,n)n+2/(diagonala)c ting on Hn(Xz,C ) (see [3,Thm. I]). In the
by the classes of these lines (cf. [4,Thm.7]). special case n = 2, this gives
The main results of this paper can be stated roughly as follows:
(i) One half of the term 48(m/2)* in the above formula of p(XL)
corresponds to the subspace of NS (XL) @ Q spanned by the classes of
curves lying on (the intersection of XL with) certain quadric surfaces of where the index set 8; =
the form
z2-c.xy=0
(up to permutation of coordinates x, y, z, w).
(ii) The term 24(m/3)* corresponds, in a similar sense, to curves lying
on cubic surfaces of the form
is naturally regarded as a subset of the character group of GL and V(a)
(resp. V(0))is the eigenspace of H2(X%,C ) with character a (resp. trivial
w3 - c .x ya = 0. character) which is known to be 1-dimensional. In particular, the Picard
number of the (complex) Fermat surface X : is
(iii) Another half of the term 48(m/2)* corresponds similarly to curves
lying on quartic surfaces of the form
Next we recall the following structure theorem of B&, which has been
formulated and partially proven in Shioda [4] and recently fully proven by
More precise statements will be given in $3. Aoki [I]. The formula (1.2) is a consequence of this result in view of (2.3).
Concerning the structure of the Ndron-Severi group of the (complex) We call an element a = (ao,a l, az, aa) E B:n decomposable if a; + aj =
Fermat surface X$,, the following problems remain to be studied: 0 (mod na) for some i # j, and indecomposable otherwise.
i) to find curves corresponding to the "exceptional" term ~(m); Theorem (8%). (i) If (m, 6) = 1, then 8; consists of decomposable
ii) to find genwators of NS (Xk) over P. elements.
4 AOKI AND SHIODA
(ii) If (m,6 ) > 1, then every indecomposable element a = (ao,a lla n, as) Observe that wc is a multiple of p,(cl.(C)) such that
of 8% with GCD(u,) = 1 i.7 equal (up to permutation) to one of the "stan-
dard" elements a,, Pi or 7 j belo<w, except for finitely many "exceptional"
elements which exist only for m 180:
as is clear from the invariance of the intersection number under an automor-
phism.
a) m = 2d, ai = (i,d -t i, m - 2i, d), 1 5 i < d, (i,d ) = 1,
b) im # = m /244. Pi = (i,d + i,d + 2i, m - li), 1 < i < dl (i,d ) = 1, ThTenh eI orreepmre 0se.n ts Fao rd eacnoym mpo s>ab l1e, leelte Lm ednetn ootfe BaL li,n ea nodf Pc3on lvyeinrsge loyn eXv:e. ry
i # m/3, m/4,m /6.
c) jm # = ;4 36d,. y j = (j,d + j, 2d + j, m - 3j), 1 5 j < dl (j,d ) = 1, (d2e.c8o)m posable element is repreWseLn.t GedL b =y s-omm e3 .l ines. Further
>
If a = (ao,a l,a2, a3) is an element of 8% with GCD(ai) = d 1, Proof. See the proof of [4, Thm. 71.
- mid.
set a: = ai/d and m' = Then a' = (ab,a:,aL,a:) is an element
Bk,
of with GCD(a:) = 1, and one checks easily that the morphism
In the next section, we shall exhibit certain curves on X& which rep-
f: X % XL, given by (xi)H (29) induces the map resent the "standard" indecomposable elcments of 8% stated in Theorem
f *: H2(x&,)-+ H2(x:) (8%). Note that, for this purpose, it sufices to consider the case a = all
PI, and 71,b ecause in general we have V(a)" = V(ta)f or any automor-
such that f *V(al)= V(a).T hus, in order to construct explicit curves on phism a of C inducing at : < H ct on the subfield of m-th roots of unity
X k corresponding to V(a),it sufices to consider the case GCD(a;)= 1. (t E (Z/m)X)w,h ile cl.(C)O = cl.(CO),C 0 being the conjugate of C urder
Now define, for any character a of G k , o (this niakes sense since our surface X: is defined over Q).
3. Main Results
Then p, is the projector of H;,.,,,(Xk) to V(a).I n particular, if a E BL
and if E = cl.(C) is the class of an algebraic cycle C on X;, then p,(E) ia Theorem 1. Let m = 2d(d > 1) and a1 = (1,d + 1, m - 2, d). Let
an ulgebraic cohomology class generating the subspace V(a)p rovided that C denote the curve of degree m in P3 defined by
# #
p,(t) 0. Note that pa(() 0 if and only if the "intersection number" + +
# (3.1) z2 - qixy = 0, xd yd G w d= 0.
pa(t).EZI 0.
Then C is a nonsingular irreducible curve lying on X L which represents
Definition. Given a E 8% and a curve C on XL, we say (for short) that the Hodge class a1 and which satisfies
C represents the IIodge class a if the following two conditions are satisfied.
Let G = G&.
Theorem 2. Let m = 3d(d 2 1) and 71 = (1,d + 1,2d + 1, m - 3).
Let C denote the curve of degree m in P3 defined by
def # #
(2.6) wc = a(y)cl.(g(C)) 0, i.e., wc -t~c0 . + +
(3.3) w3 - m x y z = 0, xd yd zd = 0.
gEG/Gc
6 AOKI AND SFIIODA
4. Proof of Theorems 1 and 2
Then C is a nonsingular irreducible curve lying on X& which represents
the IIodge class 71 and which satisfies 4
First of all look at the polynomial identities:
Theorem 3. Let m = 2d(d > 2) and = (1, d + 1, d + 2, m - 4).
Xk
(i) In case d is odd, let C denote the curve of degree 2m on defined
.
(in addhion to the Fermat equation) by Replacing x, y, . . by xd, yd, .. . , it is clear that the curve C defined by (3.1)
or (3.3) in P3 actually lies on the Ferrnat surface XL. It is easy to check
that C is nonsingular. Since a nonsingular complete intersection curve in
+ +
P3 of rnultidegree (d,e) has the genus 1 de(d e - 4)/2, we have
Then C is a nonsingular irreducible curve which represents the Hodge class
By the adjunction formula, the self-intersection number of C is given respec-
pl and which satisfies
tively by
In what follows, we shall prove only Theorem 1, since Theorem 2 can be
proven in the same way.
(ii) In case d is even, let C' denote the curve of degree 2m in P3 defined For g = [1: (1 : 52: (31 E G = G ,: let
by
+ Bk .
Then a1 = (1, d 1, m - 2, d) E is expressed as pl(g) = u(g) ~(g).
Then C' ia a singular irreducible curve lying on Xk which represents the For si~nplicity,w e set 5; = ~;(i= 1,2,3) so that ud(g) = €1, p(g) = €3
Hodge class P1 and which satisfies and ~(g=) tle3(~=; fl). Now the curve g(C) ig defined by the following
equations:
m, Hence g stabilizes C if and only if a(g) = €1 = €3 = 1. Thus
fi,
Remark. (a) The value of etc. in the defining equations of
C are fixed once and for all. (b) Theorems 1,2,3 (and Theorem 0 in 52)
x;(~)
remain valid for the Fermat surface in characteristic p, provided p
On the other hand, we see easily:
does not divide m. This will be clear from the proof.
8 AOKI AND SIIIODA
Since Gc is contained in I'Cer(al), we can define the element wc of Now we are ready to compute wc . ~jcby (2.6). Let K(a) denote the
NS(Xk) @ I C m in (2.6): image of Ker(u) in GIGc, and let K(u)* -- K(u) - {id.). Similarly we
define K(r), K(p), etc. Then
-
In order to compute wc Bc by (2.7), we need to know the intersection
numbers (C . g(C)) for g E Ker(u) U Ker(r) U I<er(p). We distinguish the
Note that a(= al) and a .ud induce nontrivial characters on each of the
three cases: (a) g E Ker(a), g @ Ker(r), (b) g @ Ker(u), g E Ker(~),
subgroups K(u), K(r), K(P), and K(T)n K(p) of G/Gc, and so we have
and (c) g @ Ker(u) U Ker(r), g E Ker(p). CK(o).
a(g) = -1, etc. Hence the above is equal to
Case (a). In this case, we have €1 = 1, €3 = -1. By (4.6), Cng(C)
consists of the 2d points PC= (1, c2,24/2~,0)(c2=d -1). For each P = PC,
let Op denote the' local ring of Xk at P, regarded as a localization of the
afline ring k[y, z, w] of Xk - {x = 0) (x = 1). The intersection multiplicity Since [G : Gc] = 2m, this implies
of C n g(C) at P is equal to the length of the quotient ring
completing the proof of Theorem 1. Q.E.D.
Therefore we have 5. Proof of Theorem 3
First we note that the curve C' in P3 defined by (3.7) actually lies on
X& in case m = 2d with d even. This follows from the elementary identity
Case (b). In this case, we have a(g) f 1, €1 = €3 = *I. The inter-
section C n g(C) consists of the 2d points P,, = (0, q, 0,l) and
- -m)
Qq = (q,0,0,1) (qd = if €1 = I, and of the d points P,, only
if €1 -1. The intersection multiplicity of C n g(C) at each P,, or Qq
is computed as before and is equal to 2. Therefore (C . g(C)) = 4d or 2d, by substituting z, . . . by xdI2,.. . . T-he) c/urave .C' has the d singular points
according as €1 = I or -1. P,, = (l,qJ0 ,O) where 9d/2 = (- 1 & To fin d the normalization
of C', observe that xy is a square on C' as the first equation of (3.7) shows.
Case (c). In thiscase, wehave&) # 1, €1 = -1 andc3 = 1. Suggested by this, we consider the following curve 6' in P4:
There are d intersection points (1,0,0, c) (cd = -1) in C n g(C), and the
multiplicity at each of them is equal to 2. Thus we have (C - g(C)) = 2d.
Since el = u ~ ( ~th)e, c ase (b) and (c) can be summarized as follows: It is easy to check that C is nonsingular and that the projection of P4
to Pn (x, y, z,w,t) I--+ (2, y, z, w) defines a birational morphism of ?I onto
Z'
C' in case d is even, and an isomorphism of to C in case d is odd. As a
10 AOKI AND SNIODA
nonsingular complete intersection curve in P4, the genus of 6 is given by Therefore, with the same notation as in the proof of Theorem 1 (54),
d .2 .2 . (d + 2 + 2 - 5)/2 + 1 = 2d2 - 2d + I. Therefore we have by the we have (write p = PI):
adjunction formula
and
[
+
pa(C1) = 2d2 - d I (pa : arithmetic genus)
(5.3)
The rest of the proof is simillar to that of Theorem 1 given in $4. We
shall sketch it only for the case d odd, omitting details.
c3]
For g = [l : (1 : 52 : E G, we set
By (2.7) and (5.5), we have
Then we have
This proves part (i) of Theorem 3. Part (ii) is similar (and even simpler),
so it will be omitted. Q.E.D.
For each g E G - Gc, the intersection of the curve C and its transform
g(C) is described as follows:
(1) Assume g &/Ker(a). (1) If g E Ker(r), there are d points (0, y, z,O),
+
yn a z d = 0, with intersection multiplicity 4. (2) If g E Ker(adr),
+
there are d points (x, 0, z, 0), xd -zd = 0, with multiplicity 4.
(3) If g E ~ e r ( oo~r )g E Ker(a2), then there are m points (x, y, O,O),
+
xm ym = 0, with multiplicity 2. (4) If g E G - Ker(ad)U Ker(u2), there
+
are m points (x, y, O,O), xm ym = 0, with multiplicity 4.
(11) Assume g E Kcr(a), g &/Ker(r). Then there are m points (x, y,O, O),
+
xm ym = 0, with multiplicity 2, and m2 points of the form (x, y, z, w),
+
xd yd = 0, ZW f_ 0, with multiplicity 1.