Abstracts - Fall 2010 

Speaker: Abhinav Kumar

Title: All the elliptic fibrations on a generic Jacobian Kummer surface

Abstract: By a theorem of Sterk, there are only finitely many elliptic fibrations on a K3 surface over the complex numbers, up to automorphism. However, explicitly computing all the elliptic fibrations is often a non-trivial problem. We describe the solution for a case of arithmetic and geometric interest: Kummer surfaces of Jacobians of genus 2 curves without extra endomorphisms, where the automorphism group is understood by work of Keum and Kondo.

Speaker: Anthony Licata

Title: Hilbert Schemes and Heisenberg Categorification

Abstract: Nakajima and Grojnowski constructed an action of a Heisenberg algebra on the cohomology of Hilbert schemes of points on an ALE space. We construct an action of a categorified Heisenberg algebra on the derived categories of coherent sheaves on these Hilbert schemes. This is joint work with Sabin Cautis.

Speaker: Kefeng Liu

Title: Positivity and vanishing theorems for ample vector bundles

Abstract: By using differential geometric method, we derive several new positivity and vanishing results for ample vector bundles on projective manifolds. This is joint work with Xiaokui Yang and Xiaofeng Sun.

Speaker: Ziv Ran

Title: Boundary modifications of Hodge bundles and enumerative geometry

Abstract: In the classical enumerative geometry of smooth curves, degeneracy loci involving Hodge bundles and Brill-Noether maps have played an important role. Extending these methods to stable curves runs into some issues, e.g, excess degeneracy at the boundary. Here we describe an approach to dealing with these issues in the case of schemes of degree 2, resulting in a presentation of the locus of stable hyperelliptic curves (with extra data), plus a manageable residual locus, as a degeneracy locus of a boundary modification of the Hodge bundle. The methods involve a particular blowup of the Hilbert scheme, on which the the appropriate bundle modification can be constructed. Unlike some Gromov-Witten methods, they are applicable beyond curves of compact type; in fact it's the noncompact-type curves that necessitate the blowing up.