Arithmetic and Geometry II

245 Pages·2004·13.35 MB·English

Progress in Mathematics
Vol. 36
Edited by
J. Coates and
S. Helgason
Arithmetic and
Geometry
Papers Dedicated to
I.R. Shafarevich on the Occasion
of His Sixtieth Birthday
Volume I1 Geometry
Michael Artin,
John Tate,
editors
Birkhauser Birkhauser
Boston. Basel. Stuttgart Boston Basel Stuttgart
Editors:
Michael Artin John Tate
Mathematics Department Mathematics Department
Massachusetts Institute of Technology Harvard University
Igor Rostislavovich Shafarevich has made outstanding contribu-
Cambridge, MA 02139 Cambridge, MA 02138
tions in number theory, algebra, and algebraic geometry. The flour-
ishing of these fields in Moscow since World War I1 owes much to his
I
This book was typeset at Stanford University using the TEX document preparation influence. We hope these papers, collected for his sixtieth birthday,
system and computer modern type fonts by Y. Kitajima. Special thanks go to will indicate to him the great respect and admiration which mathema-
Donald E. Knuth for the use of this system and for his personal attention in the de- ticians throughout the world have for him.
velopment of additional fonts required for these volumes. In addition, we extend
Michael Artin
thanks to the contributors and editors for their patience and gracious help with im-
Igor Dolgachev
plementing this system.
John Tate
A.N. Todorov
Library of Congress Cataloging in Publication Data
Main entry under title:
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
Geometry 1 Michael Artin ; John Tate, ed. -
Boston ; Basel ; Stuttgart : Birkhauser, 1983.
(Arithmetic and geometry ; Vol. 2) (Progress
in mathematics ; Vol. 36)
ISBN 3-7643-3133-X
NE: Artin, Michael (Hrsg.); 2. GT
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-
cal, photocopying, recording or otherwise, without prior permission of the copy-
right owner.
Birkhauser Boston, Inc., 1983
@
ISBN 3-7643-3133-X
Printed in USA
I.R. Shafarevich, Moscow, 1978
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. Hinebruch, Arrangements 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 Certain 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 PGSp4 of Curves
M. Raynaud, Courbes sur une variCtC abklienne 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
Some Algebro-Geometrical Aspects
of the Newton Attraction Theory*
V. I. Arnold
To I. R. Shafnrevich
According to the Zcldovich theory1, the observed large scale structure
of the universe (the drastically non-uniforrri distribution of galaxy clusters)
is explained by thc geometry of caustics of a mapping of a Lagrange sub-
manifold of the synlplectic total space of the cotarlgent bundle to its base
spncr. This Lagrar~ges ubmanifold is fornled by the particle velocities.
Coriternporary theory of the hot universe predicts a smooth potential veloc-
ity fie!d at an early stage (when the universe was about 1000 times "s~rdler"
1l~;inr~ow ), At this stage thc Lagrange manifold is a cotangent bundle sec-
tion. '['hen it, evolves according to IIarr~iltoniane quations of motion, and
hcncc coirtirlues to be Lagrangian. Ilowcvcr, it clops not need to be a scc-
tion at all times. l'hc sel of critical values of its projeclion on the base
spac:c is called the caus~ic. At the caustic the particle density becomes
i:lfiuik (mat,henlatically); the cnuslic is the place whrre clustering occurs
(genc.r:~tiono f galaxies and so on).
l'hc singularities of caustics and thcir mctamorphoscs are classified in
a w:ly usunl for the I,agrarlgian singularity theory: Ak,D k,E k,. . . . This
is true for norl-interacting parlicles or for particlcs in any potential field.
liut iu caws wt~errt he field is gcnernted by the particlcs, a new tlificulty
occurs. Attcr tllc caustic has been formed the force ficld is no longrr
smooth (bec;twc of density singularities). tlencc our Iagrange rrlanifold
rnq :~ccpires ingul:~ritics. Thus we are Icd to the problcrrl: to gcncralize
t,t~c:1 ,agrnnge mappings singularity theory to the casc where the 1,agrarige
'frolrt a letter to M. Artin, May 10, 1982
'~rnold,V .I., Shandarin, S.F., Xrldovich, Ya. R., the Large Scale Stxucture of the
Universe I, Geophys, Astrophys. Fluid Dynamics 1982.
Arnold, \'.I., Surgery of singularities in potential collisionless rncdia and caustics rneta-
morphoses in 3-space, 'l'rudy Seminary, 3, (1982), 22-58.
2 V.I. ARNOLD NEWTON ATTRACTION THEORY 3
source manifold becomes a Lagrange variety. component charges of different signs. Such a hyperbola does not attract
points of the convex parts of the plane it bounds.
[Lagrange varieties occur also in other situations, for i~~stancine the
study of the shortest length function on a nianifold with boundary (see V.I. Now let us consider any real algebraic hypersurface. We call a point
Arnold, Lagrange varieties singularities, asyrnpototical rays and the open p hyperbolic with respect to thc hypersurface if all real lines through p
swallowtail, Funkt. Anal. 15 (1982) 1-14). In this case typical singularities meet the hypersurface at real points only. The following generalization of
Newton's theorem holds:
of Lagrange varieties are ttiosc of the set or odd degree polynoniials having
a root of multiplicity greater lhan one half of the degree. (The symplec-
tic structure of the space on polynomials is inherited from the invariant (2) The charge associated to a hyperbolic point does not attract this
symplectic structure on the space of binary forms.) Hut Lagrange variety point.
singularities arising from the Newton attraction theory seem to be different
(and in any case they are not known).] For instance, let us consider a plane algebraic curve of even degree,
consisting of a sequence of ovals one inside other (their number being equal
The first step in the study of such Lagrange variety singularities is the
to one half of the degree). Let the charges change signs from each oval to
study of the singularities of force fields, gcnerated by clustering of free
the next one and let the density he natural. Such a curve does not attract
particles. Among typical clusterir~gsi n the pl;tne, elliptic and hyperbolic
,!I4 singularities occur at some (exceptional) rnorncnts in time. At these sin- points inside the inner oval.
The theorem on the attraction of exterior points leads to results in the
gularities, the particle density is inversely proportional to a quadratic form
geometry of confocal quadrics which seem to be ncw. The generalization to
(in suitable local coordinates). Thus constant density lines are homothetic
ellipses or hyperbolas. We arc led to the problcrn of Newtonian attraction noriconnccted hypcrboloids is direct; for instance, the attraclion of points
by an rlliptic or hyperbolic layer. between the components of the hyprrbola charged as above does not change
The theory of attraction by elliptic layers is classic. The first results are if we substitute for the hyperbola a larger confoci~ol ne. For quadratic forms
of other signatures one needs to consider dilkreritial forms of appropriate
due to Newton: a uniform spherical layer does not attract interior points,
degree instead of charges and instcad of the Ncwton-Coulomb attraction
and attracts ext,rrior ones as if the mass were concentrated at its center.
These results were extended to the case of ellipsoids by Ivory, but the law one needs to consider the Biot-Savart law and its higher dimensional
hyperbolic cast seems not to be settled by classical authors. arialogues.
Two generalizations of the Newton iriterior points theorem arc formu-
lated below. Let us consider a level hypersurfitce of a real polynomial in a
euclidean space of dimensiori h, and a point not on the hypersurface. We Rcceived June 2, 1982
call natural the density inversrly proportion;il to tlie gradient length. Let us
distribute a Coulomb chargc (force inverscly proportiod to the (n - 1)st Professor V. I. Arnold
power of the tlistancc) along the hypersurface, with natural density but MaLheniatics Department
+
with a sign, depending on the choscn point: for points of the hypcrsur- Moscow State University
+
face which one can see from the choscn one, - for those obstructed once, Moscow 117 234
for thosc obstructed twice, arid so on. We call such a charge "associated" USSR
to the chosen point. Then tlie following generalized Newton theorem holds:
(1) The charge on a second degree hypersurface, associated to a point,
docs not attract this point.
For instance, let us consider a plnrie hyperbola with natural density and
Smoothing of a Ring Homomorphism
Along a Section
M. Artin and J. Denef
To I.R. Shnfarevich
Inti-od~~ction:T his paper studics the problem of srrioolhing a homomor-
phism of corrlrnutative rings along a section. The data needed to pose the
problem make up a corn~nutativcd iagram
of :~Ifirlcx l\emes, wch thal Y is finitely prcscnted ovrr X. Our standard
not,ni,ion is that X, X,Y arc thr spectra of A, 71, Il respcctivcly, arid
lhat Ij is ;L firiitcly prescr~ted/ l-algebra. (In the body of the text, we work
primarily with thr rings ralhcr t!lan with (heir spectra. This reverscs the
arrows.) 'l't~e problrxn is to ernbed the commutative diagram (0.1) inlo a
larger one,
such that
(i) a is smooth, and
(ii) q5 is srnooth whercvcr possible - roughly speaking, except above the
singular (nonsrnooth) locus of n.
6 ARTIN AND DENEB SMOOTHING OF A RING IIOMOMORPHISM 7
It is probable that a diagram (0.2) exists under fairly general conditions, There is a finitely presented B-algebra C and an A-homomorphism
perhaps whenever the map W --+ X is regular. Special cases have been u0: C --, A/a:
known for some time (N6ron [13], IClkik [9], I'fister [ll]). We prove its
existence for normal X in two special cases: when X (section 2),
-1
and when A, 7f are henselian local rings with the same completion, and a
is sniooth at every point of s(W) except the closed point - the isolated such that
singularity case (section 3).
Scction 4 contains a versior~o f N6ron1s p-desingularisation [13], and in
section 5 we apply the results to henselian local rings of dimension 2. (i) qh = CY is smooth.
The subject of this paper was motivated by the following observation. (ii) Let p E Spec @/a) - V(D). Then q5 is smooth at u0-l (p) E Spec C.
Let A, a be a henselian pair, the completion of A, X = Spec A, and = (iii) u0q5 = so (modulo~).
spcc2. Suppose that every diagram (0.1) can be embedded in a diagram
(0.2) satisfying (i) and (ii). Then A has the approximation property, i.e., Note that if the pair (A,R)i s henselian, the homomorphism u0 can be
every system of polynomial equations over A with a solution in A has a extended to a section a: C+ A.
solution in A. It is klmwn [2] that the hcnselization of R[xl,. . . ,x n] at
(P, 21,. . . , s,) has the approximatiori property if R is an excellent discrete Proof of Theorem (1.1). We first corisider the ho~nogeneousl inear
valuation ring with prime p. It is not known whrthcr the henselization of
case, with yo = 0, a 0, and with f(y) rcplaccd by
k[[xl,.. . , X,]][X,+~, . . . ,x,] at (xl,. . . , x<,) has the approxi~nationp roperty, =;
though this is true when r = 2 and rL 5 [14, 41.
We want to acknowlcdge the help of D. Popescu. Part of the work in this
paper was done in collaboration with him, and is an~iouncedin [3, see also
To avoid confusion, we denote the ring defined by these equations by B'.
171. We would also like to thank 11. Eismbud for explaining the resolution
In this case the scheme Y' = SpecB' represents, functorially, the kernel
(1.11) to us.
of the linear map defined by J, i.e., there is an exact sequence of group
schemes over X = Spec A:
1. The Complete Intersection Case
An obvious choice of smoothing of Y' can bc obtained by mapping a free
This section refines a method of Tougeron (18, 21, which can be applied
Y' J
when U is a relative complete int,crsection over A. As always, module to = ker(An 4 A'):
B = A[y]/(/) is a finilcly presented A-nlgcbra, with y = (yl,.. . , y,) and
f = (jl:. . .,j ,)t. WC fix sonic yo E An, and suppose r 5 n. Let the
Jacobian matrix at yo be J = (af,/~yj)(yo),a nd let I) denote the ideal
generated by the r-rowcd (maxirrial) minors of J. If JP = 0, the corresponding map
Theorem (1.1). Assume that /(yo) = 0 (modulo D~R),w here R is
some ideal of A. Thus y = yo defines un A-homomorphism
will send 6,NX to Y'. So we may take C = A[zl,.. . , z,], and Spcc C =
G&. Conditior~s(i ), (iii) of Lttc theorem are trivially vcrilicd.
SMOOTIlING OF A RING HOMOMORPHISM 9
Lemma (1.7). With the above notation, condition (ii) of theorem These relations determine the resolution for a generic matrix J, which
(1. I) holds if the sequence (1.5) iu exact at all points p of X at which J has can be written as
maximal rank, i.e., p $Z V(D).
Proof of Lemma (1.7). Condition (ii) only concerns points of % =
SpccC lying over such 1). So, by localizing, we may assume that J has Here P is contraction by the vector dt E A"An*. If w = (w,) represents an
niaximal rank. Then the maps (l.4), (1.5) determined by J arc surjective, element of W = Ar+lAn and y = (yi) a vector in An, then P is given as
IJ' is a projective A-module, and ./r is smooth. Clearly 6 2 +P Y' will be
snlooth if and only if AN +P y' is surjective. Namely, if P is surjective then
it splits because Y' is projective, and so AN == Z' @ y' with Z1p rojective.
Siir~ilarly,6 ,Nx cz %' X xY1.S ince Z' is projective, %I is smooth over X, The relations (1.10) imply that this is a complex, and it is proved in [8] that
hence P is also smooth. Conversely, if Ggx -+ Y' is smooth, one proves the complex is exact when J is a generic matrix. Therefore, for generic J,
surjectivity by the Nnkayama Lemma. it is split exact (i.e., J has a section) locally at any point at which J has
maximal rank (is surjective), and by pull-back, it is also split exact a1 such
It follows from this lemma that we can get a solution in the hornogcneous a point for any J. By Lemma (1.7), the associated map (1.5) satisfies the
linear case by choosing 'F so that the sequence (1.5) is exact. 11owcver, we requiren~erltso f Theorem (1.1).
prcfcr to rriakr a canonical choice, one which corrcsponds to a resolution For the nonlinear case, wc will need some extra variables to absorb higher
of the A-module M = AT/J when J is a generic matrix. The resolution in order terms, and so we embed W into An@ATAnb y the rule w -t v = (vjp),
that case is trrated in [8]. with
I,eL at3 denote thc rntries of J. An r-rowed minor of J is determined by
an r-multi-index p = (jl, . . . ,j ,), with 1 < jl < . . . < jT 5 n. We denote
this rainor by d,:
where a = {j)U p, arranged in increasing order, arid where the sign is
Thus D is the ideal generated by the set {d,,). 1,et d = (d,) denote the row (-I)", if j = j, in a. Thus W is defined in An @ ArAn by the equa-
tions L,:
vcctor obtaird by some ordering of the multi-indices p.
Therc are natural relatioris among {aij) and {d,,), as follows: Let a
denotr an (I. + I)-multi-index a = (jl,. . . ,jT+1)w ith
for all (r -1 1)-multi-indices a. The map 'I is i~iducedb y
and let a, denote the r-multi-index obtained by deleting j,:
Expansion by rriinorv leads to the relations
T+1
(1.10) (- 1)" aiiy d,,, = 0. With this notation, the ring required by Theorem (1.1) is pre~ent~eady
v=l Cv = A[v]/(LV),w lirrc I,, is the systc~n( 1.14) of linear equations, and the
10 AltTIN AND DENEF
map 4': B'-+ C, is defined by the substitution (1.15). The linear cquation wherc Q,,,,,, are polynomials in {ujp) having no tern1 of degree < 2.
has thus become Si~lccd p is a minor of J, there is a matrix Np with
(1.22) d,, 1 = JNp (1t hc r X r identity).
which is implicd by the syslem L, (1.14).
This allows us to factor out J on the lefl from the right-hand term of (1.21),
If one wants to avoid the rcferencc to [8], onc can check the rcquircmcnts
using thc index p'. We can also collect thc terrns d,, into a factor dt on the
of Lemma (1.7) for the system L, directly, as follows: Suppose that some
minor d = don is invertible, say pn = (1,. . . , r). Then equations (1.3) right.
Y'
can be used to establish an iso~rlorphisrno f with thc frcc module with
coordinates {y,+l, . . . , y,). Let Y" be a free n~odulew ith coordinates
{Y:+l, . . . , &), and define a ho~nornorphismS t:Y " -, W by
where E,,, is the matrix ( E ~ I ~e~tc.) L~et, ~ ,
Thc relations L, (1.14) determine vjp uniquely in tcrms of y;., if j 5 r.
Y" Y'
Thc map P'S': --t is This is a rnatrix whose row and column indices are (j,p). Thus
where v = (vj,), and
Therefore PIS' is surjective, hcncc P' is surjective, as required. This
completes our discussion of the linear case.
Now consider the general case. Since f(yn) E D2a, we may write
In this way the original equation is related to thc homogencous linear system
(1.3).
Let C, denote t3hc quotient of A[u] by thc ideal dcfincd by the linear
+
relatio~is( I. 14) with v = u q (1.26):
for some clements c,,tp, of a. Substitute
-
Then C,, is a B-algebra, via (1.20). because of (1.16) and (1.25). The sub-
wlicrc u -- (ujp)a nd d = (d,,). Taylor's expansion gives us stitution u = 0 implies v = u+q O(rnoduloa), by (1.21) and (1.19), hence
clcterrnincs ari A-homomorphism a,: C, + Ala. Notr that the Jacobian
matrix of the systerr~( 1.26) is congrucril to the identity (rnodulo(u, a)). Let
7 denote the Jacobian of this system. Let