Algebraic Geometry Seminar
Ana Cristina Lopez Martin
University of Salamanca
Moduli spaces of semistable sheaves on singular genus 1 curves
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.
Friday September 12, 2008 at 4:00 PM in SEO 636