# Reconstructing toric quiver flag varieties from a tilting bundle

## TL;DR

This paper demonstrates that toric quiver flag varieties can be reconstructed as moduli spaces of modules over endomorphism algebras of tilting bundles, generalizing the reconstruction of projective spaces.

## Contribution

It establishes that every toric quiver flag variety is isomorphic to a moduli space of cyclic modules over a specific algebra, extending known results about projective spaces.

## Key findings

- Toric quiver flag varieties are isomorphic to moduli spaces of modules.
- Generalizes the reconstruction of projective spaces from endomorphism algebras.
- Provides a new perspective on the structure of toric quiver flag varieties.

## Abstract

We prove that every toric quiver flag variety $Y$ is isomorphic to a fine moduli space of cyclic modules over the algebra $\text{End}(T)$ for some tilting bundle $T$ on $Y$. This generalises the well known fact that $\mathbb{P}^n$ can be recovered from the endomorphism algebra of $\bigoplus_{0\leq i\leq n} \mathcal{O}_{\mathbb{P}^n}(i)$.