Sommersemester 2008

15.04.2008  Alex Küronya (Budapest Univ. of Technology/Univ. Essen)
Ampleness and asymptotic cohomology
Abstract: One of the fundamental facts about the cohomology of sheaves on a projective variety is Serre's criterion for ampleness. It gives a characterization of ample invertible sheaves in terms of vanishing of higher cohomology of sheaves twisted by a high enough multiple of the line bundle in question. Originally conjectured by Lazarsfeld, and eventually proved in a joint effort with de Fernex and Lazarsfeld we provide a similar characterization of ampleness via asymptotic cohomology.
22.04.2008 Gerard van der Geer (Universiteit van Amsterdam)
Cycle relations on Jacobian varieties
Abstract: By using the Grothendieck-Riemann-Roch theorem we derive cycle relations modulo algebraic equivalence in the Jacobian of a curve. The relations generalize the relations found by Colombo and van Geemen and are analogous to but simpler than the relations recently found by Herbaut. I will also talk about the conjectured structure of the 'tautological' ring. This is joint work with Alexis Kouvidakis
29.04.2008 Margarida Melo (Universita Roma 3)
Compactified Picard stacks over the moduli space of stable curves
Abstract: For a projective curve X of genus g, the generalized jacobian J(X) is compact if and only if X is a curve of compact type. There are several constructions of compactifications of J(X), differing from one another in various aspects such as the geometric interpretation or functorial properties. In this talk we will explain how to construct geometrically meaningful Artin stacks \bar{P}_{d,g} over the moduli stack of stable curves, giving a functorial compactification of the degree d Picard variety for families of stable curves. The connection with Caporaso's compactification of the degree d universal Picard variety will also be discussed.
06.05.2008 Walter Gubler (Univ. Dortmund/HU Berlin)
Canonical measures on abelian varieties
Abstract: Chambert-Loir has introduced measures on the Berkovich analytic space associated to a projective variety which are analogues of the top-dimensional wedge-product of first Chern forms. In the talk, these measures will be described for a subvariety of an abelian variety.
13.05.2008Alessandro Verra (Universita Roma 3)
On the Prym moduli spaces in low genus
Abstract: The unirationality problem is discussed for some moduli spaces related to the Prym moduli spaces R_g, in low genus. For g < 8 the unirationality result is obtained via the study of the Hilbert scheme of (canonical) curves of genus g in a cubic hypersurface defined by a symmetric 3 x 3 determinant of linear forms.
20.05.2008 Marian Aprodu (IMAR Bucharest)
Koszul cohomology and geometry of complex curves
Abstract: We discuss recent advances on two conjectures that relate numerical invariants of curves to Koszul cohomology with values in suitable line bundles. These conjectures were made in the 80s by Green, and Green-Lazarsfeld respectively.
27.05.2008 Gavril Farkas (HU Berlin)
The Koszul geometry of the moduli space of curves
Abstract: Given a moduli space, what is the "best" effective divisor one can write down on it? We present a very general method to construct divisors on moduli spaces using the syzygies of the parameterized objects. Applications of this method include a doubly infinite string of counterexamples to the Harris-Morrison Slope Conjecture and a vastly simpler rederivation of all the calculations of Harris, Mumford and Eisenbud of divisor classes on moduli spaces.
02.06.2008S. Ramanan (Chennai Math. Institute)
An algebraic geometric understanding of fundamental groups through Higgs pairs
03.06.2008 Paul Larsen (HU Berlin)
The Mori cone of the moduli space of pointed, rational curves
Abstract: Results of Gibney, Keel and Morrison show that the Mori cone of curves for the moduli space of n-pointed, genus g curves is entirely determined by the geometry of the moduli space of (g+n)-pointed rational curves, $\bar{M}_{0, g+n}$. We present work on the cone of curves of $\bar{M}_{0,n}$ for small n, including a partial proof of Fulton's conjecture about nef divisors on $\bar{M}_{0,7}$.
17.06.2008 Milena Hering (Univ. of Connecticut)
The moduli space of n points on the projective line and the Koszul property
Abstract: The moduli space of weighted points on the projective line is a GIT quotient (\mathbb{P}^1)^n)//SL(2,\mathbb{C}) which admits a natural embedding into projective space. I will present a formula for the Hilbert function of this embedding, and I will show that the second Veronese embedding is Koszul. This is joint work with Benjamin Howard.
17.06.2008 Mihnea Popa (Univ. of Illinois at Chicago)
Numerical inequalities via syzygies and generic vanishing
Abstract: I will explain how a refinement of the Green-Lazarsfeld Generic Vanishing Theorem, general Fourier-Mukai transform theory, and the Syzygy Theorem of Evans-Griffith, can be used to prove a higher dimensional analogue of the classical Castelnuovo-de Franchis inequality for surfaces. I will then use this inequality to provide some bounds on the irregularity of special classes of projective varieties. Joint work with Giuseppe Pareschi.
15.07.2008 Ana-Maria Castravet (Univ. of Arizona)
Hypergraphic divisors, curves, and morphisms
Abstract: The birational geometry of the moduli space of stable, n-pointed, rational curves is still largely unknown. For example, it is not known if the cone of effective divisors is polyhedral, if the cone of effective curves is generated by 1-strata (Fulton's conjecture), or if it has only finitely many small modifications. In joint work with Jenia Tevelev, we find that some of this complexity can be explained by the fact that this moduli space can be interpreted as the Brill-Noether locus of some very special reducible curves related to hypergraphs. Using this we construct many new extremal divisors and birational morphisms with unexpected exceptional loci.