Blow ups of $\mathbb{P^n}$ as quiver moduli for exceptional collections

Abstract

Suppose $P^n_m$ is the blow up of $\mathbb{P}^n$ at a linear subspace of dimension $m$, $\mathcal{L}=\{L_1,\ldots,L_r\}$ is a (not necessarily full) strong exceptional collection of line bundles on $P^n_m$. Let $Q$ be the quiver associated to this collection. One might wonder when is $P^n_m$ 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 achieve such isomorphism using $\mathcal{L}$ of length $3$. As a result, $P^n_m$ is the moduli space of representations of a very simple quiver. Moreover, we realize the blow up as morphism of moduli spaces for the same quiver.