# Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

## TL;DR

This paper investigates the moduli space of stable sheaves supported on genus three curves within a smooth quadric surface, demonstrating its rationality and computing its Betti numbers through a detailed geometric analysis.

## Contribution

It provides a comprehensive description of the moduli space, including its rationality, Betti numbers, classification of sheaves, and a global geometric construction involving flips and blow-downs.

## Key findings

- The moduli space is rational.
- Betti numbers are explicitly computed.
- Stable sheaves are classified via resolutions and extensions.

## Abstract

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of alpha-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.