Wintersemester 2007/08
| 23.10.2007 | Katharina Ludwig (Universität Hannover) |
| Singularities of the moduli space of Prym curves | |
| Abstract:The moduli space \overline{R}_g of Prym curves is a compactification of the moduli space R_g of pairs (C,L) of a smooth curve of genus g and a square root L of the trivial bundle of C. In my talk I will explain the local analytic structure of \overline{R}_g and use this description to determine the locus of non-canonical singularities. I will then show that every pluricanonical form on the regular locus of \overline{R}_g extends holomorphically to a desingularisation of \overline{R}_g, in the same style as J. Harris and D. Mumford did for the moduli space of curves. | |
| 31.10.2007 | Herbert Lange (Univ. Erlangen) |
| Prym-Tyurin varieties | |
| Abstract:A Prym-Tyurin variety (PTV) of exponent q is an abelian subvariety P of a Jacobian variety (JC,Theta) such that the restriction of the canonical polarization Theta to P is the q-fold of a principal polarization. PTV's of exponent 1 are Jacobians, PTV's of exponent 2 are classical Prym varieties. Not many PTV's of exponent >= 3 were known until recently. After a discussion of the known results on the subject I will outline a new construction which produces many new examples for any exponent. This is a report on two recent papers on Prym-Tyurin varieties (joint work with V.Kanev as well as A. Carocca, R.Rodriguez and A.Rojas). | |
| 06.11.2007 | Bernd Sturmfels (Univ. of California at Berkeley) |
| Elimination Theory for Tropical Varieties | |
| Abstract:Elimination theory is aimed at computing the image of an algebraic variety under a polynomial map. In this lecture we present work with Jenia Tevelev and Josephine Yu on a new approach to elimination theory in the piecewise-linear setting of tropical algebraic geometry. | |
| 13.11.2007 | Filippo Viviani (HU Berlin) |
| Cohomological support loci for the ideal of an Abel-Prym curve | |
| Abstract:Cohomological support loci were introduced by Green-Lazarsfeld to prove some generic vanishing theorems for irregular varieties. Later, Pareschi-Popa used them to define a concept of regularity on principally polarized abelian varieties which resembles many of the property of the Castelnuovo-Mumford regularity on projective spaces. As an application, they studied the cohomological support loci of the Abel-Jacobi curve on a Jacobian, obtaining a characterization of them. In this talk, based on a joint work with S. Casalaina-Martin and M. Lahoz, we consider the analougous problem for the "next class of curves", namely Abel-Prym curves inside Prym varieties. | |
| 20.11.2007 | Grigory Mikhalkin (Univ. of Toronto) |
| Real Enumerative Geometry | |
| Abstract:The talk will be a survey of currently known enumerative invariants computing the number of real curves and techniques for their computation. | |
| 04.12.2007 | Gavril Farkas (HU Berlin) |
| The Kodaira dimension of the moduli space of Prym varieties | |
| Abstract: The best understood abelian varieties are Jacobians of curves. It is a notoriously hard problem to write down the general abelian variety of given dimension and one can obtain a larger family than Jacobians by considering Prym varieties associated to etale double covers of curves. We prove that the moduli spaces of Prym varieties of dimension greater than 13 are of general type. This is in contrast with results of Clemens, Donagi and Verra stating that for Prym varieties of dimension at most 6, the moduli space is unirational. | |
| 15.01.2008 | Klaus Hulek (Universität Hannover) |
| Moduli of polarized symplectic manifolds | |
| Abstract: In many ways irreducible symplectic manifolds behave similar to K3-surfaces, although it is known that the global Torelli theorem fails in general. Nevertheless it is possible to relate moduli spaces of polarized irreducible symplectic manifolds to quotients of type IV domains by an arithmetic group. We analyse the situation and prove that certain moduli spaces of polarized irreducible symplectic manifolds are of general type. This is joint work with V. Gritsenko and G.K. Sankaran. | |
| 29.01.2008 | Norbert Hoffmann (FU Berlin) |
| On moduli of bundles on a curve | |
| Abstract: Two classical results about coarse moduli schemes of vector bundles on an algebraic curve, namely their rationality (due to King-Schofield) and the non-existence of Poincare families (due to Ramanan), are related via the corresponding moduli stacks. This allows to generalize them to vector bundles with extra structures. Finally, I'll also report on joint work in progress with I. Biswas, which partially generalizes these results to principal bundles, using the Picard group of their moduli stacks. | |
| 05.02.2008 | Bernd Sturmfels (Univ. of California at Berkeley) |
| Powers of Linear Forms | |
| Abstract: What is the dimension of the space of polynomials of a given degree that are annihilated by given powers of fixed vector fields? We aim to express this dimension as a piecewise quasipolynomial of the powers and the degree. This talk presents an approach to this problem using toric degenerations of the Cox-Nagata ring of the blowup of a finite set of points in projective space. Our main example, the case of eight or fewer general points in the plane, takes us on a combinatorial journey to del Pezzo surfaces and their moduli. Work in progess with Zhiqiang Xu. | |
| 12.02.2008 | Angela Ortega (Univ. Essen/Morelia) |
| Dolgachev's conjecture on the moduli space of rank 3 bundles | |
| Abstract: For a curve of genus 2, the moduli space of rank 3 vector bundles is a double cover of the 8-dimensional projective space branched along a sextic hypersurface. Coble proved a century ago that there exists a unique cubic hypersurface in 8-space which is singular along the Jacobian variety of the curve and invariant under the natural action coming from the torsion points of the Jacobian. Dolgachev has made the striking conjecture than these two seemingly completely unrelated hypersurfaces are in fact projectively dual to each other. I will discuss the background of the problem and present a proof of Dolgachev's conjecture. | |
| 13.02.2008 | Pierre Schapira (University of Paris VI) |
| From D-modules to deformation quantization modules | |
| Abstract:
I will explain how the study of the sheaf of rings
D_M on a complex manifold M naturally leads to the
construction of the sheaf E_X of microdifferential operators
on the cotangent bundle X=T^*M, then to the notion of
*-algebras and more generally to that of DQ-algebroids
on a complex Poisson manifold X. The classical finiteness and duality theorems of Cartan, Serre and Grauert for O-modules have their counterparts for DQ-modules and I will discuss a kind of Riemann-Roch theorem for such modules. This is a joint work with Masaki Kashiwara. |