Moduli of sheaves supported on curves of genus two in a quadric surface

TL;DR

This paper investigates the moduli space of stable sheaves supported on genus two curves in a quadric surface, demonstrating its rationality and classifying the sheaves through resolutions and extensions.

Contribution

It provides a classification of stable sheaves on genus two curves in a quadric surface and proves the rationality of their moduli space.

Findings

The moduli space is rational.

Classification of stable sheaves via resolutions and extensions.

Betti numbers computed through alpha-stability variations.

Abstract

We study the moduli space of stable sheaves of Euler characteristic 1, supported on curves of arithmetic genus 2 contained in a smooth quadric surface. We show that this moduli space is rational. We give a classification of the stable sheaves involving locally free resolutions or extensions. We compute the Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs.