Vector bundles with a fixed determinant on an irreducible nodal curve

Abstract

Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank $r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar{L}}$ be the closure of its subset consisting of GPBs with fixed determinant $\bar{L}$. We define a moduli functor for which $M_{\bar{L}}$ is the coarse moduli scheme. Using the correspondence between GPBs on $X$ and torsion-free sheaves on a nodal curve $Y$ of which $X$ is a desingularization, we show that $M_{\bar{L}}$ can be regarded as the compactified moduli scheme of vector bundles on $Y$ with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves on $Y$. The relation to Seshadri--Nagaraj conjecture is studied.

7 pages

References in corpus (2)

• Degeneration of moduli spaces and generalized theta functions
• Moduli spaces of SL(r)-bundles on singular irreducible curves