# Birational geometry of the moduli space of pure sheaves on quadric surface

## TL;DR

This paper investigates the birational geometry of a specific moduli space of stable sheaves on a quadric surface, describing its structure through a sequence of blow-ups and blow-downs, and relating it to a projective bundle over a Grassmannian.

## Contribution

It provides a detailed birational description of the moduli space of stable sheaves with given invariants on a quadric surface, including explicit geometric transformations.

## Key findings

- Explicit birational map constructed between the moduli space and a projective bundle
- Description of the moduli space via smooth blow-ups and blow-downs
- Connection established with a Grassmannian-based projective bundle

## Abstract

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.