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

485 Pages·1983·6.32 MB·English

by  Professor V. I. Arnold (auth.)

Progress in Mathematics
Vol. 36
Edited by
J. Coates and
S. Helgason
Springer Science+Business Media, LLC
Arithmetic and
Geometry
Papers Dedicated to
I.R. Shafarevich on the Occasion
of His Sixtieth Birthday
Volume II Geometry
Michael Artin,
John Tate,
editors
1983 Springer Science+Business Media, LLC
Editors:
Michael Arlin John Tate
Mathematics Department Mathematics Department
Massachusetts Institute of Technology Harvard University
Cambridge, MA 02139 Cambridge, MA 02138
This book was typeset at Stanford University using the TEX document preparation
system and computer modern type fonts by Y. Kitajima. Special thanks go to
Donald E. Knuth for the use of this system and for his personal attention in the de
velopment of additional fonts required for these volumes. In addition, we extend
thanks to the contributors and editors for their patience and gracious help with im
plementing this system.
Library of Congress Cataloging in Publication Data
Main entry under title:
Arithmetic and geometry.
(Progress in mathematics; v. 35-36)
Contents: v. I. Arithmetic- v. 2. Geometry.
l. 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-
II. Arlin, Michael. III. Tate, John Torrence,
1925- . IV. Series: Progress in mathematics
(Cambridge, Mass.) ; v. 35-36.
QA7.A67 1983 513'.132 83-7124
ISBN 978-0-8176-3133-8
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Geometry I Michael Artin ; John Tate, ed. -
(Arithmetic and geometry; Vol. 2) (Progress
in mathematics; Vol. 36)
ISBN 978-0-8176-3133-8 ISBN 978-1-4757-9286-7 (eBook)
DOI 10.1007/978-1-4757-9286-7
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.
© Springer Science+Business Media New York 1983
Originally published by Birkhauser Boston in 1983
ISBN 978-0-8176-3133-8
Igor Rostislavovich Shafarevich has made outstanding contribu
tions in number theory, algebra, and algebraic geometry. The flour
ishing of these fields in Moscow since World War II owes much to his
influence. We hope these papers, collected for his sixtieth birthday,
will indicate to him the great respect and admiration which mathema
ticians throughout the world have for him.
Michael Artin
Igor Dolgachev
John Tate
A.N. Todorov
-··
~~
/lhJ-~
~3~
p~~~
Volume II Geometry
V. 1. Arnold, Some Algebro-Geometrical Aspects of the Newton 1
Attraction Theory
M. Artin and J. Denef, Smoothing of a Ring Homomorphism Along a 5
Section
M.F. Atiyah and A.N. Pressley, Convexity and Loop Groups 33
H. Bass, The Jacobian Conjecture and Inverse Degrees 65
R. Bryant and P. Griffiths, Some Observations on the Infinitesimal 77
Period Relations for Regular Threefolds with Trivial Canonical Bundle
H. Hironaka, On Nash Blowing-Up 103
F. Hirzebmch, Arrangements of Lines and Algebraic Surfaces 113
V.G. Kac and D.H. Peterson, Regular Functions on Certain Infinite- 141
dimensional Groups
W.E. Lang, Examples of Surfaces of General Type with Vector Fields 167
Yu.I. Manin, Flag Superspaces and Supersymmetric Yang-Mills 175
Equations
B. Moishezon, Algebraic Surfaces and the Arithmetic of Braids, I 199
D. Mumford, Towards an Enumerative Geometry of the Moduli Space 271
of Curves
C. Musili and C.S. Seshadri, Schubert Varieties and the Variety of 327
Complexes
A. Ogus, A Crystalline Torelli Theorem for Supersingular K3 Surfaces 361
M. Reid, Decomposition of Toric Morphisms 395
M. Spivakovsky, A Solution to Hironaka's Polyhedra Game 419
A.N. Tjurin, On the Superpositions of Mathematical Instantons 433
A.N. Todorov, How Many Kahler Metrics Has a K3 Surface? 451
0. Zariski, On the Problem of Irreducibility of the Algebraic System 465
of Irreducible Plane Curves of a Given Order and Having a Given
Number of Nodes
Some Algebro-Geornetrical Aspects
of the Newton Attraction Theory*
V.I. Arnold
To I. R. Shafarevich
According to the Zcldovich theory1, the observed large scale structure
of the universe (the drastically non-uniform distribution of galaxy clusters)
is explained by the geometry of caustics of a mapping of a Lagrange sub
manifold of the symplectic total space of the cotangent bundle to its base
space. This Lagrange submanifold is formed by the particle velocities.
Contempor£J.ry theory of the hot universe prediets a smooth potential veloc
ity field at an early stage (when the universe was about 1000 times "smaller"
than now). At this stage the Lagrange manifold is a cotangent bundle sec
tion. Then it, evolves according to Hamiltonian equations of motion, and
hence continues to be Lagrangian. However, it does not nt!ed to be a sec
tion at all times. The set of critical values of its projedion on the base
space is called the caus!;ic. At the caustic the particle density becomes
infiuite (mathematically); the caustic is the place where clustering occurs
(g;ener:.tiou of galaxies and so on).
The singularities of caustics and their metamorphoses are classified in
a w:1.y usual for the Lagrangian singularity theory: Ak,Dk,Ek,· ... This
is true for non-interacting particles or for particles in any potential field.
But iu cases where the Held is generated by the particles, a new difficulty
occurs. After the caustic has been formed the force field is no longer
smooth (because of density singularities). Hence our Lagrange manifold
may aequire singularities. Thus we are led to the problem: to generalize
the Lagrange mappings singularity theory to the ease where the Lagrange
•rrom a letter to M. Artin, May 10, 1982
1 Arnold, V.I., Shandarin, S.F ., Zeldovich, Ya. B., the Large Scale St.ructure of the
Universe I, Geophys, Astrophys. Fluid Dynamics 1982.
Arnold, V.I., Surgery of singularities in pot<'ntial collision less media and caustics meta
morphoses in 3-spacc, Trudy Seminary, 3, (J982), 22-58.
2 V.I. ARNOLD
source manifold becomes a Lagrange variety.
[Lagrange varieties occur also in other situations, for instance in the
study of the shortest length function on a manifold with boundary {see V.I.
Arnold, Lagrange varieties singularities, a.'lympototical rays and the open
swallowtail, Funkt. Anal. 15 {1982) 1-14). In this case typical singularities
of Lagrange varieties are those of the set of odd degree polynomials having
a root of multiplicity greater than one half of the degree. (The symplec
tic structure of the space on polynomials is inherited from the invariant
symplectic structure on the space of binary forms.) But Lagrange variety
singularities arising from the Newton attraction theory seem to be different
(and in any case they are not known).]
The first step in the study of such Lagrange variety singularities is the
study of the singularities of force fields, generated by clustering of free
particles. Among typical clusterings in the plane, elliptic and hyperbolic
D singularities occur at some {exceptional) moments in time. At these sin
4
gularities, the particle density is inversely proportional to a quadratic form
(in suitable local coordinates). Thus constant density lines are homothetic
ellipses or hyperbolas. We are led to the problem of Newtonian attraction
by an elliptic or hyperbolic layer.
The theory of attraction by elliptic layers is classic. The first results are
due to Newton: a uniform spherical layer does not attract interior points,
and attracts exterior ones as if the mass were concentrated at its center.
These results were extended to the case of ellipsoids by Ivory, but the
hyperbolic case seems not to be settled by classical authors.
Two generalizations of the Newton interior points theorem are formu
lated below. Let us consider a level hypersurface of a real polynomial in a
euclidean space of dimension h, and a point not on the hypersurface. We
call natural the density inversely proportional to the gradient length. Let us
distribute a Coulomb charge {force inversely proportional to the (n- l)st
power of the distance) along the hypersurface, with natural density but
with a sign, depending on the chosen point: + for points or the hypcrsur
face which one can see from the chosen one, - for those obstructed once, +
for those obstructed twice, and so on. We call such a charge "associated"
to the chosen point. Then the following generalized Newton theorem holds:
(1) The charge on a second degree hypersurface, associated to a point,
does not attract this point.
For instance, let us consider a plane hyperbola with natural density and