LIST OF SPEAKERS WITH TITLES AND ABSTRACTS

"Journées de géométrie algébrique"

in Lille from 18 to 20 May, 2005.

Marian Aprodu (Bayreuth) « Syzygies of d-gonal curves »

Abstract.

We apply a degenerate version of a result due to Hirschowitz, Ramanan
and Voisin to verify Green and Green-Lazarsfeld conjectures over explicit
open sets inside each d-gonal stratum of curves X with d<[g(X)/2]+2.
By the same method, we verify the Green-Lazarsfeld conjecture for
any curve of odd genus and maximal gonality. The proof invokes
Voisin's solution to the generic Green conjecture as a key argument.

Indranil Biswas (TIFR, Mumbai) «On the stability of the tangent bundle of a hypersurface in a Fano variety of Picard number one»

Abstract.
This is a joint work with Georg Schumacher. Let M be a smooth projective complex Fano variety with Pic(M) =Z such that the tangent bundle TM is semistable. Let Z in M be a smooth hypersurface of degree strictly greater than deg(TM)(dim_CZ−1)/(2dim_CZ−1) and satisfying the condition that the inclusion of Z in M gives an isomorphism of Picard groups. Then we show that the tangent bundle of Z is stable. A similar result holds for smooth complete intersections in M.

Olivier Debarre (Strasbourg) « On coverings of simple abelian varieties »

Abstract.
To any finite covering f:Y--> X of degree d between smooth complex projective manifolds, one associates a vector bundle E_f of rank d-1 on X whose total space contains Y. Lazarsfeld showed that E_f is ample when X is a projective space. We show an analogous result when X is a simple abelian variety and f does not factor through any nontrivial isogeny X'--> X. This result is obtained by showing that E_f is M-regular in the sense of Pareschi--Popa, and that any M-regular sheaf is ample.

Valery Gritsenko (Lille) « Mumford-Hirzebruch proportionality principle and the moduli spaces of K3 surfaces »

Abstract. The moduli space M(2d) of polarized K3 surfaces of degree 2d
is the quotient of the complex homogeneous domain of dimension 19
by a subgroup of an integral orthogonal group of signature (2,19).
The moduli spaces M(2d) are unirational if d is between 1 and 11
or d=17, 19 (Mukai).
If d=p^2, where p is a prime number, then M(2d) is of general type
if p goes to infinity (Kondo). An effective bound for the prime p is not
known.

In this talk I give a review of my joint result
with K. Hulek and G. Sankaran about the Kodaira dimension of M(2d).
Using the Mumford-Hirzebruch proportionality principle
and special automorphic forms on the orthogonal group O(2,19)
we can find an effective bound for polarization $d$
for which the moduli spaces M(2d) are of general type.

Laurent Gruson (Versailles) « The Wronskian of two binary forms »

Abstract. For any degree d the wronskian W is a finite flat morphism from
the grassmannian G of lines in the projective space of binary forms of degree d
to the projective space P of binary forms of degree 2d-2 . We study the
first Boardman loci of W in order to prove that the discriminant of W is
integral (this question was raised by Verdier and settled by Meulien).

Joseph Le Potier (Jussieu) «Cohomologie du schéma de Hilbert ponctuel d'une surface : travaux de M. Haiman, G. Danila et L. Scala» Résumé

Manfred Lehn (Mainz) « Singular symplectic moduli spaces II »

Abstract.
(Part I: Sorger, Part II Lehn, joint work with D. Kaledin) Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of the class of moduli spaces found by O'Grady. As a consequence, singular moduli spaces that do not belong to this exceptional class do not admit symplectic resolutions.

Werner Nahm (Dublin) «Integrable quantum field theories, algebraic K-theory and Yangians»

Abstract.

Conformally invariant quantum field theories in two dimensions have
been accepted as mathematical systems (for example under the name of
vertex operator algebras). Their integrable deformations have not yet
been explored by mathematicians, but it is clear that they have even
richer structures. Some related conjectures concern the torsion part
of K₃ for the complex numbers, sum forms for modular functions which
generalise the Rogers-Ramanujan identities, and character formulas
for representations of Yangians.

Viacheslav V. Nikulin (Liverpool) «Correspondences of a K3 surface with itself via a general Mukai vector»

Abstract.

A primitive Mukai vector v=(r,H,s) with 2rs=H2 defines
the moduli Y of sheaves on a K3 surface X where Y is again a K3
surface. Divisorial conditions on moduli of X which imply that
Y\cong X were studied in series of papers alg-geom/0206158, 0304415,
0307355, 0309348. But for a general Mukai vector v (alg-geom/0309348)
the results were very complicated and unsatisfactory. Recently (see second
variant of alg-geom/0309348) I significantly simplified these results and
obtained satisfactory results in general. This will be the talk about.

Chris Peters (Grenoble) «Motivic polynomials for mixed Hodge structures»

Abstract.

This is a preliminary report of work in progress with Jozef 
Steenbrink.  Mixed Hodge structures in geometry come from  sheaf complexes
 equipped with two filtrations, the weight and the Hodge filtrations. To
these  we associate a certain  integer polynomial which  behaves well
under various  natural operations. These properties  make it possible to
calculate rather  easily certain Hodge numbers or Euler-characteristics
defined from these.  Some examples coming from singularity-theory will be
provided.

Miles Reid (Warwick) «K3s and Fano 3-folds»

Abstract.

The general program for classifying Q-K3 surfaces and Q-Fano 3-folds,
together with recent progress. (Compare [S. Altınok, G. Brown and
M. Reid, Fano 3-folds, K3 surfaces and graded rings, in Topology and
geometry: commemorating SISTAG (National Univ. of Singapore, 2001), Ed.
A. J. Berrick and others, Contemp. Math. 314, AMS, 2002, pp. 25--53,
preprint math.AG/0202092, 29 pp.]).

Cregory Sankaran (Bath) «The Prym moduli space R₅»

Abstract.

In some ways, the moduli space R₅ of etale double covers of genus 5
curves have not been as well studied as other Prym moduli spaces. We
indicate some gaps, and describe attempts to fill them.

N.I. Shepherd-Barron (Cambridge) «Non-equivariant deformations»

Abstract.

We exhibit an action of an algebraic group G on a
0-dimensional complex scheme X where there is no natural
action of G, or its formal group, on any miniversal deformation of X.
(Joint with Ekedahl.)

Cristoph Sorger (Nantes) « Singular symplectic moduli spaces I »

Katrin Wendland (Warwick) « On a family of smooth algebraic K3 surfaces and their associated CFTs »

Abstract. 
The moduli space of Einstein metrics on K3 is well-known to algebraic 
and differential geometers. Physicists have introduced the notion of 
conformal field theories (CFTs) associated to K3, and the moduli space of 
these objects is well understood as well. It can be interpreted as a
generalisation of the moduli space of Einstein metrics on K3, which allows 
us to introduce this space without having to use background knowledge from 
conformal field theory.
However, just as no smooth Einstein metrics on K3 are known explicitly, 
the explicit construction of CFTs associated to K3 in general remains an 
open problem. The only known constructions which allow to deal with 
families of CFTs use orbifold techniques and thus give CFTs associated to 
K3 surfaces with orbifold singularities.
We use classical results by Shioda and Inose along with the known 
structure of the respective moduli spaces to explicitly construct a family 
of CFTs which are associated to a family of smooth algebraic K3 surfaces. 
Although these CFTs were known before, it is quite remarkable that they 
allow a description in terms of a family of smooth surfaces whose complex 
structure is deformed while all other geometric data remain fixed.