# Linear data for framed sheaves via exceptional sequences

## TL;DR

This paper describes the moduli spaces of framed sheaves on a smooth projective surface using linear algebraic data derived from exceptional sequences, connecting geometric moduli to quiver representations.

## Contribution

It provides a new method to realize moduli spaces of framed sheaves as principal bundles over quiver representation stacks, under certain conditions.

## Key findings

- Moduli spaces are described via linear data.
- Connection established between sheaves and quiver representations.
- Framework applies to surfaces with full strong exceptional sequences.

## Abstract

Let $X$ be a smooth projective surface with a full strong exceptional sequence $\mathfrak{E}$. Under certain conditions, we describe the moduli spaces of framed sheaves on a line in $X$ via linear data, i.e. by realizing them as principal bundles over a stack of representations of the bound quiver associated to $\mathfrak{E}$.