On the geometry of the moduli space of sheaves supported on curves of genus two in a quadric surface

TL;DR

This paper investigates the structure of the moduli space of stable sheaves supported on genus two curves in a quadric surface, demonstrating its rationality, computing Betti numbers, and classifying stable sheaves via resolutions.

Contribution

It provides the first detailed analysis of the moduli space's geometry, including rationality proof, Betti number computation, and classification of stable sheaves.

Findings

The moduli space is rational.

Betti numbers of the moduli space are explicitly computed.

Stable sheaves are classified through locally free resolutions.

Abstract

We study the moduli space of stable sheaves of Euler characteristic 2, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers and we give a classification of the stable sheaves involving locally free resolutions.