The Picard group of the moduli space of sheaves on a quadric surface

TL;DR

This paper investigates the structure of the Picard group of the moduli space of semistable sheaves on a smooth quadric surface, revealing new behaviors for small discriminant cases using geometric invariant theory.

Contribution

It introduces new methods for analyzing the Picard group of moduli spaces on quadric surfaces, especially for small discriminant sheaves, through resolutions and GIT techniques.

Findings

New descriptions of the Picard group for small discriminant cases

Identification of novel behaviors in moduli spaces of sheaves

Application of geometric invariant theory to moduli space analysis

Abstract

In this paper, we study the Picard group of the moduli space of semistable sheaves on a smooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class. We focus especially on moduli spaces of sheaves of small discriminant, where we observe new and interesting behavior. Our method relies on constructing certain resolutions for semistable sheaves and applying techniques of geometric invariant theory to the resulting families of sheaves.