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

Abstract

Suppose $P_m^n$ is the blow up of ${\mathbb{P}}^n$ at a linear subspace of dimension $m$, $\mathcal{L}=\{L_1,\dots ,L_r\}$ is a (not necessarily full) strong exceptional collection of line bundles on $P_m^n$. Let $Q$ be the quiver associated to this collection. One might wonder when is $P_m^n$ the moduli space of representations of $Q$ with dimension vector $(1,\dots ,1)$ 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, $P_m^n$ 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.

13 pages, comments welcome. Minor change in introduction. arXiv admin note: text overlap with arXiv:1803.06533