Ana Cristina Lopez Martin : Moduli spaces of semistable sheaves on singular genus 1 curves

Posted by Mihnea Popa , part of the Algebraic Geometry Seminar.

At

Sept. 12, 2008, 4 p.m.

In

SEO 636

Abstract

Atiyah's characterization of vector bundles allowed Tu to give a geometric description of the moduli spaces of semistable sheaves on smooth elliptic curves. All those results can be obtained in a very simple way as an application of the Fourier-Mukai transform on an elliptic curve. In this talk, we will consider the case of some degenerations of elliptic curves, focussing on Kodaira fibers of type $E_N$. For a cycle $E_N$ of projective lines, we will show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves O(-1) supported on one irreducible component. The Fourier-Mukai transform allows then to prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.