The Picard group of a coarse moduli space of vector bundles in positive characteristic

TL;DR

This paper investigates the Picard group of a coarse moduli space of semistable vector bundles over a curve in positive characteristic, establishing its structure and properties despite challenges posed by positive characteristic.

Contribution

It proves that the Picard group of the moduli space is isomorphic to Z, identifies its ample generator, and shows the space is locally factorial in positive characteristic, extending known results from characteristic zero.

Findings

Picard group of the moduli space is Z

Identified the ample generator of the Picard group

Proved the moduli space is locally factorial

Abstract

Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M_{r,L}^{ss} denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M_{r,L}^{ss}) = Z, identify the ample generator, and deduce that M_{r,L}^{ss} is locally factorial. In characteristic zero, this has already been proved by Dr\'{e}zet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive caracteristic.