Algebraic Geometry and its Applications: Proceedings of the 8th Algebraic Geometry Conference, Yaroslavl’ 1992. A Publication from the Steklov Institute of Mathematics. Adviser: Armen Sergeev

263 Pages·1994·5.24 MB·English

by  Walter L. Baily Jr. (auth.)

Alexander Tikhomirov
Andrej Tyurin (Eds.)
Algebraic Geometry
and its Applications
Aspect~f
Mathematic~
Edited by Klas Diederich
Vol. E 2: M. Knebusch/M. Kolster: Wittrings
Vol. E 3: G. Hector/U. Hirsch: Introduction to the Geometry of Foliations, Part B
Vol. E 5: P. Stiller: Automorphic Forms and the Picard Number of an Elliptic Surface
Vol. E 6: G. Faltings/G. Wustholz et al.: Rational Points*
Vol. E 7: W. Stoll: Value Distribution Theory for Meromorphic Maps
Vol. E 9: A. Howard/P.-M. Wong (Eds.): Contribution to Several Complex Variables
Vol. E 10: A. J Tromba (Ed.): Seminar of New Results in Nonlinear Partial Differential
Equations*
Vol. E 13: Y. Andre: G-Functions and Geometry*
Vol. E 14: U. Cegrell: Capacities in Complex Analysis
Vol. E 15: J.-P. Serre: lectures on the Mordell-Weil Theorem
Vol. E 16: K. Iwasaki/H. Kimura/S. Shimomura/M. Yoshida: From Gauss to Painleve
Vol. E 17: K. Diederich (Ed.): Complex Analysis
Vol. E 18: W. W. J Hulsbergen: Conjectures in Arithmetic Algebraic Geometry
Vol. E 19: R. Racke: lectures on Nonlinear Evolution Equations
Vol. E 20: F. Hirzebruch, Th. Berger, R. Jung: Manifolds and Modular Forms*
Vol. E 21: H. Fujimoto: Value Distribution Theory of the Gauss Map of Minimal
Surfaces in Rm
Vol. E 22: D. V. Anosov/A. A. Bolibruch: The Riemann-Hilbert Problem
Vol. E 23: A. P. Fordy/J C. Wood (Eds.): Harmonic Maps and Integrable Systems
Vol. E 24: D. S. Alexander: A History of Complex Dynamics
Vol. E 25: A. Tikhomirov/A . Tyurin (Eds.): Algebraic Geometry and its Applications
* A Publication of the Max-Planck-Institut fur Mathematik, Bonn
Volumes of the German-language subseries "Aspekte der Mathematik" are listed at the end of the book.
Alexander Tikhomirov
Andrej Tyurin
(Eds. )
Algebraic Geollletry
anel its Applications
Proceedings of the 8th Algebraic
Geometry Conference,
Yaroslavl' 1992
A Publication from the
Steklov Institute of Mathematics
Adviser: Armen Sergeev
Alexander Tikhomirov Andrej Tyurin
Department of Mathematics Steklov Mathematical Institute
State Pedagogical Institute Vavilova 42
Respublikanskayastr. 108 Moscow 117966
Yaroslavl' 150000 Russia
Russia
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Algebraic geometry and its applications: proceeedings of
the 8th Algebraic Geometry Conference, Yaroslavl', 1992;
a publ. of the Steklov Institute of Mathematics 1
Alexander Tikhomirov; Andrej Tyurin (ed.). -
Braunschweig; Wiesbaden: Vieweg, 1994
(Aspects of mathematics: E; Vol. 25)
NE: Tichomirov, Aleksandr S. [Hrsg.]; Algebraic Geometry
Conference (8,1992, Jaroslavl'); Matematireskij Institut Imeni
V. A. Steklova (Moskva); Aspects of Mathematics 1 E
Mathematics Subject Classification:
14C05, 14C30, 14025, 14E05, 14E07, 14F05, 14F25, 14F45, 14Jxx, 14110, 81Exx, 10Dxx
ISBN 978-3-322-99344-1 ISBN 978-3-322-99342-7 (eBook)
DOI 10.1007/978-3-322-99342-7
All rights reserved
© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1994
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Foreword
This volume consists of articles presented as talks at the Algebraic Geometry Conference
held in the State Pedagogical Institute of Yaroslavl'from August 10 to 14, 1992. These
conferences in Yaroslavl' have become traditional in the former USSR, now in Russia,
since January 1979, and are held at least every two years. The present conference, the
eighth one, was the first in which several foreign mathematicians participated. From the
Russian side, 36 specialists in algebraic geometry and related fields (invariant theory,
topology of manifolds, theory of categories, mathematical physics etc.) were present. As
well modern directions in algebraic geometry, such as the theory of exceptional bundles
and helices on algebraic varieties, moduli of vector bundles on algebraic surfaces with
applications to Donaldson's theory, geometry of Hilbert schemes of points, twistor spaces
and applications to string theory, as more traditional areas, such as birational geometry of
manifolds, adjunction theory, Hodge theory, problems of rationality in the invariant theory,
topology of complex algebraic varieties and others were represented in the lectures of the
conference.
In the following we will give a brief sketch of the contents of the volume.
In the paper of W. L. Baily three problems of algebro-geometric nature are posed. They
are connected with hermitian symmetric tube domains. In particular, the 27-dimensional
tube domain 'Fe is treated, on which a certain real form of E7 acts, which contains a "nice"
r
arithmetic subgroup e, as observed earlier by W. Baily. The author discusses an approach
to finding the interpretation of 'Fe /re as a moduli space of a certain family of polarized
algebraic varieties. The approach is based on the relation between Severi varieties of F. Zak
and irreducible symmetric tube domains of lR-rank 3.
The paper ofM. C. Beltrametti, G. M. Besana and A. 1. Sommese concerns the dimension
of the adjoint linear system K x Q9 Vj9( n-2} for quadric fibrations. Namely, let (X", LII)
be a smooth n-dimensional projective manifold with a very ample line bundle LII on it and
let (X, L) be its reduction (in the natural sense that there exists a blow up r : XII ---+ X
of a projective manifold X at a finite set B such that L = (r*LII)** is ample and LII =
r* L Q9 OX" (-r-1 (B)). (It is known from the adjunction theory that except for an explicit
list of special pairs (X II, LII) this reduction exists and is unique up to an isomorphism.)
Using the technique, developed by M. C. Beltrametti and A. J. Sommese and the earlier
results of G. M. Besana, the authors prove the following main result of this paper (theorems
2.2 and 3.3): if (X, L) is a quadric fibration over a surface, then hO(Kx" Q9 LII(n-2}) ~ 2
with 3 possible exceptiones, described explicitely.
VI Foreword
D. Butler considers in his paper the left transform ME of a given vector bundle E with a
slope I-l(E) ;::: 2, generated by global sections, on the curve e, this left transform being
defined in a standard way as the kernel of the evaluation map HO(E) i8I Oc ---+ E.
The main result of the paper (theorem 1) states that the (semi)stability of E implies the
(semi)stability of its left transform ME.
The paper of H. d'Souza concerns the description of the intermediate Jacobian J(X) of a
threefold X with a Del Pezzo fibration via Prym-Tyurin variety P associated with a cylinder
map for a family of lines on X. The main result of the paper gives the condition under
which the principally polarized abelian varieties (J(X), 8) and (P,3) are isomorphic,
where 8 is the Poincare theta-divisor on J(X) and 3 the canonical polarization of P: this
is the condition that the incidence correspondence for the family of lines on X has no fixed
points. The work is essentially based on the results and methods developed by V. Kanev for
the description of (P, 3) of the Del Pezzo fibrations via Prym-Tyurin variety.
The ramification, decomposition and inertia subgroups of the Cremona groups are studied
in the paper of M. H. Gizatullin. The author shows that the simplicity of the Cremona group
of the plane is a consequence of a positive solution of a congruence subgroup problem. A
representation of Manin's group of minimal cubic surfaces in the Cremona group of the
space is given. A nontriviality of some ramification groups is established.
In the first part of the paper of A. L. Gorodentsev a survey of helix theory is given. This theory
was developed in order to obtain constructive description of the set of exceptional vector
bundles (i.e., bundles E with dim Hom(E, E) = 1 and Exti(E, E) = 0 for i > 0). In the
second part the author discusses some connections between helix theory and arithmetical
properties of nonsymmetrical bilinear forms on lattices and formulates some conjectures
about braid group action on the set of semiorthogonal bases of these forms. In the third part
there are given examples of calculations with these forms and their groups of isometries.
The moduli space MJ2] of curves of genus 3 (plane quartics) with a fixed point of order
2 (or, equivalently, with a fixed unramified double covering) is considered in the paper
of P. I. Katsylo. This space is closely connected with the moduli space M3 of curves of
genus 3, the classical problem of rationality of which was recently solved affirmatively by
Katsylo. In this paper the similar result for MJ2] is proved: MJ2] is rational variety. The
proof is based on the reduction of the problem to that of rationality of fields of invariants
for certian representations of SL2, stated by the author earlier.
The moduli space en(m) of stable weightened ordered n-tuples of points in ]pI under
the natural action of PGL2(<C) is studied in the paper of A. A. Klyachko (here m =
(ml' m2, ... , mn) is the vector of weights). As it was shown earlier by the author, this
space naturally arises in the description of the structure of rank-2 stable sheaves with odd
determinant on ]p2. In the paper the original method of the study of en (m) is introduced,
based on the interpretation of this space (more precisely, of its natural compactification by
a finite number of points) in terms of the geometry of the spatial polygons in the Euclidean
3-space. This interpretation enables the author to compute the Betti numbers of en (m) and
give the explicit description of en (m) for small n.
In the paper of S. A. Kuleshov vector bundles E with the condition Extl(E, E) = 0
Foreword VII
on Del Pezzo surfaces are studied. These bundles are called rigid in view of the lack of
their infinitesimal deformations. The author proves that any such bundle is a direct sum of
exceptional bundles, i.e. rigid and simple (EndE = iC), these last being indecomposable.
The fundamental group G = 71"1 (C2 - D) of the complement of an algebraic curve D in
C2 is treated in the paper of Vic. S. Kulikov. Certain properties of G and of the Alexander
polynomial ~D (t) of the curve D are stated, e.g., it is proved that: if D is irreducible then the
= =
commutator G' [G, G] of G is finitely generated and·rk(G' /[G', G'])/ree deg~D(t)
is an even number; if D is connected, then all the roots of ~D(t) are the roots of unit; if G
=
is an irreducible C-group, then ~D(l) ±1. Besides, the description of ~D(t) in terms
of the C-corepresentation of G is given.
In the paper of V. V. Nikulin the group 2Br(X) of elements of order 2 in the Brauer
group Br(X) of the projective algebraic variety X over IR is considered. Let s be the
number of connected components of X(IR). The author proves that the canonical map
2Br(X) -- (Z/2)S is epimorphic if H3(X(iC)/G;Z/2) = 0 and X(IR) i:- 0, where
=
G Gal(C/IR). This result is then applied to Enriques surfaces.
In the paper of A. D. Popov and A. G. Sergeev there is presented the geometric quantization
scheme for the bosonic string theory in twistor terms. Starting from the loop space of a
Lie group they define a symplectic twistor bundle over the loop space and reformulate the
geometric quantization problem in terms of this bundle. For the standard bosonic string
they recover in this way the well known critical dimension condition.
In his list of problems of complex and differentiable varieties F. Hirzebruch (1954) mentio
ned the classification of all compact complex manifolds V with b2(V) = 1 containing the
open analytic subset A such that V \ A is biholomorphic to en. The complete answer to this
problem at the moment is known for n ~ 3. The paper of Yu. G. Prokhorov deals with such
compactifications (V, A) of C4 when V is projective. The main result of the paper (theorem
3.1) states that in the case of index 3 (i.e. when K v = -3H for some H E PicV) there
exists only four such compactifications. For all of these cases V appears to be the linear
section of the grassmanian G(2, 5) in 1P9, and they differ only by the divisor A explicitely
described in each case. (Note that the case of index ~ 4, contaning projective 4-space and
quadric for V, is an easy consequence of the results of Kobayashi and Ochiai.)
It was supposed by A. Gorodentsev and A. Rudakov that for any two exceptional sheaves
on a projective plane only one ext-group can be different from zero, which was proved later
by A. Bondal and A. Gorodentsev. In the first paper of A. N. Rudakov the similar question is
studied for exceptional sheaves on a quadric surface. The result is that this is not longer true,
there are exceptional sheaves A, B on a quadric such that Hom ( A, B) and Ext 1 (A, B) are
nonzero. But it is proved at the same time that for a subclass of symmetrical exceptional
sheaves on a quadric surface the statement holds true: if both A and B are symmetrical,
then only one ext-group can be different from zero.
The aim of the second paper of A. N. Rudakov is to summarize known properties of
exceptional sheaves and vector bundles on a Del Pezzo surface and to express a set of
hypotheses about these sheaves. The main known fact about an individual exceptional
sheaf is that if the sheaf has no torsion, then it is a stable vector bundle, and if it has a
VIII Foreword
torsion, then its support is an exceptional curve. But it is worth to study not the individual
sheaves but exceptional systems of exceptional sheaves. Here the main fact is that one
can define a braid group action on the set of exceptional systems of a given length. It is
especially important to study this action when the length is maximal possible for a given
surface. In the paper there is a series of conjectures on the properties of this action.
Computation of Segre (respectively, Chern) classes of standard vector bundles on Hilbert
scheme of points on a surface, being itself an interesting problem of enumerative geometry,
attracted new attention after the recent works of A. N. Tyurin, who interpreted these classes,
in particular, top Segre classes Od, as new invariants of smooth structure of a fourfold
underlying the complex algebraic surface. In the paper of A. S. Tikhomirov it is proved that
Od is the polynomial of degree d (= degree of O-cycles on a surface S) of four invariants
x = (D2), Y = (D· Ks), z = 82(S), w = (K~), where D is a given divisor on S. As
an example, the formula for 03(X, y, z, w) is given, which coincides with the formula of
P. Le Barz obtained earlier in different terms.
In the next paper of A. S. Tikhomirov and T. L. Troshina the original formula for
04(X, y, Z, w) is found. The method used here is based on the application of the double
point formula for the immersion ("generally") of the chordal variety of a surface S in the
IDI D.
space ]p'lO by an appropriate linear subseries of for ample divisor
In the last paper of the volume A. N. Tyurin studies the modification of the Gieseker
closure of the moduli space MH (2; Cl, C2) of H-stable rank-2 vector bundles on the surface
S, with given Chern classes Cl, C2, when the polarization H, lying inside the Kahler
cone K+ of S, deforms in such a way that it passes through the "walls of chambers",
these walls being by the definition orthogonal to the vectors e E Pic( S) with conditions
cI -
e == Cl (mod2), 4C2 ::; (e2) ::; O. For the description ofthis modification the method of
geometric approximation (GA-procedure) is developed. This procedure enables the author
to compute the almost canonical spin-polynomials of S via their geometric approximation.
The correction terms of this procedure are treated, and several problems and relations with
Donaldson's theory are discussed.
We are very grateful to Tatyana Troshina, and also to Andrej Kazusev and Igor Khomutinni
kov, for doing the largest part of the technical work in the preparation of these proceedings
for publication.
Alexander Tikhomirov
Andrej Tyurin
Authors. addresses
W.L.Baily Jr., Department of Mathematics, Y.Y.Nikulin, Steklov Mathematical Institute,
The University of Chicago, 5734 Univer Vavilova 42, 117966 Moscow, GSP-l, RUS
sity Avenue, Chicago, Illinois 60637, USA SIA - [email protected]
- [email protected]
A.D.Popov, Steklov Mathematical Institute,
M.C.Beltrametti, Dipartimento di Matema ul. Vavilova 42, 117966 Moscow, GSP-l,
tica, Universita di Genova, Via L.B.Alberti RUSSIA
4, 16132 Genova, ITALY
Y.G.Prokhorov, Chair of Algebra, Depart
G.M.Besana, Department of Mathematics, ment of Mathematics, Moscow State Univer
University of Notre Dame, Notre Dame, In sity, 117234 Moscow, RUSSIA
diana 46556, USA
A.N.Rudakov, Institute for System Analysis,
D.C.Butler, Department of Mathematics, Russian Academy of Science, ul. Avtoza
University of Michigan, Ann Arbor, MI vodskaya, 23, 119280 Moscow, RUSSIA -
48106, USA [email protected]
Harry D'Souza, Department of Mathematics, A.G.Sergeev, Steklov Mathematical Institute,
University of Michigan, Flint, MI 48502- Vavilova 42, 117966 Moscow, GSP-l, RUS
2186, USA [email protected] SIA - [email protected]
M.H.Gizatullin, Department of Mathematics, AJ.Sommese, Department of Mathematics,
Samara State University, 443011, Samara, University of Notre Dame, Notre Dame, In
academician Pavlov str. 1, RUSSIA diana 46556, USA
A.L.Gorodentsev, Steklov Mathematical In A.S.Tikhomirov, Department of Mathema
stitute, Vavilova 42, 117966 Moscow, GSP-l, tics, State Pedagogical Institute ofYaroslavl',
RUSSIA Respublikanskaya 108, 150000 Yaroslavl',
RUSSIA - [email protected]
P.I.Katsylo, Moscow Institute of Electronic
Machinery, Moscow, RUSSIA T.L.Troshina, Department of Mathematics,
State Pedagogical Institute of Yaroslavl',
A.A.Klyachko, Department of Mathematics,
Respublikanskaya 108, 150000 Yaroslavl',
Samara State University, Pavlova 1,443011
RUSSIA
Samara, RUSSIA
A.N.Tyurin, Steklov Mathematical Institute,
S.A.Kuleshov, Moscow Independent Univer
Vavilova 42, 117966 Moscow, GSP-l, RUS
sity, Moscow, RUSSIA
SIA - [email protected]
Y.S.Kulikov, Moscow Institute of Engineers,
of Transport, Moscow, RUSSIA
Contents
Three Problems on an Exceptional Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Walter L. Baily, Jr.
On the Dimension of the Adjoint Linear System for Quadric Fibrations . . . . . . . . . 9
M. C.Beltrametti, G.M.Besana and A.J.Sommese
On the Stability of ME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
David C. Butler
On a Class of Del Pezzo Fiber Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Harry D'Souza
The Decomposition, Inertia and Ramification Groups in Birational Geometry. . . 39
M.H. Gizatullin
Helix Theory and Nonsymmetrical Bilinear Forms. . . . . . .. .. . . . . . . . . .. . . .. . . . 47
A.L. Gorodentsev
On the Unramified 2-covers of the Curves of
Genus 3................................................................. 61
P.I.Katsylo
Spatial Polygons and Stable Configurations of Points in the Projective Line. . . . . 67
Alexander A. Klyachko
Rigid Sheaves on Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
S.A.Kuleshov
The Alexander Polynomials of Algebraic Curves
in C2 •••••...•..•.....•.......••.....•••.•••••••••.•••••••••••••••••••.• 105
Vic. S.Kulikov
On the Brauer Group of Real Algebraic Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Viacheslav V. Nikulin