# Moduli of quiver representations for exceptional collections on surfaces

## TL;DR

This paper demonstrates that for del Pezzo surfaces, the surface can be realized as a moduli space of quiver representations associated with a full strong exceptional collection of line bundles, and explores how blow-ups relate to modifications of these collections.

## Contribution

It establishes the isomorphism between del Pezzo surfaces and moduli spaces of quiver representations, and links blow-ups to augmentations of exceptional collections.

## Key findings

- Del Pezzo surfaces are moduli spaces of quiver representations.
- Blow-ups can be recovered via augmentation of exceptional collections.
- The correspondence holds for specific stability conditions.

## Abstract

Suppose $S$ is a smooth projective surface over an algebraically closed field $k$, $\mathcal{L}=\{L_1,\ldots,L_n\}$ is a full strong exceptional collection of line bundles on $S$. Let $Q$ be the quiver associated to this collection. One might hope that $S$ is the moduli space of representations of $Q$ with dimension vector $(1,\ldots,1)$ for a suitably chosen stability condition $\theta$: $S\cong M_\theta$. In this paper, we show that this is the case for del Pezzo surfaces. Furthermore, we show the blow-up at a point can be recovered from an augmentation of exceptional collections (in the sense of L. Hille and M.Perling) via morphism between moduli of quiver representations.