# On the geometry of the moduli space of sheaves supported on curves of genus four contained in a quadric surface

## TL;DR

This paper investigates the structure of the moduli space of stable sheaves supported on genus four curves within a quadric surface, demonstrating its rationality and computing its Betti numbers.

## Contribution

It establishes the rationality of the moduli space and calculates its Betti numbers using the variation of alpha-semi-stable pairs.

## Key findings

- The moduli space is rational.
- Betti numbers are explicitly computed.
- Variation of alpha-stability is used in the analysis.

## Abstract

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of bidegree (3, 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.