# Discriminants of stable rank two sheaves on some general type surfaces

## TL;DR

This paper establishes sharp bounds on the discriminants of stable rank two sheaves on certain surfaces, utilizing tilt stability and derived category techniques, and proves the Bogomolov inequality in this context.

## Contribution

It introduces new bounds on discriminants of stable sheaves and extends the Bogomolov inequality to all characteristics for these surfaces.

## Key findings

- Sharp bounds on discriminants of stable rank two sheaves
- Description of surfaces as moduli spaces of vector bundles
- Proof of Bogomolov inequality in all characteristics

## Abstract

We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality for semistable rank two sheaves on integral surfaces in three-dimensional projective space in all characteristics.