Full Exceptional Sequence for a Fine Quiver Moduli Space

TL;DR

This paper constructs a full exceptional sequence for the derived category of a specific quiver moduli space, linking its geometry to derived category theory using advanced algebraic geometry techniques.

Contribution

It provides the first explicit construction of a full exceptional sequence for the derived category of a fine quiver moduli space of the 3-Kronecker quiver.

Findings

Explicit exceptional sequence constructed for the moduli space

Demonstrated fullness via mutations and covering arguments

Connected the geometry of the space to derived category structures

Abstract

We consider the fine quiver moduli space of representations of the 3-Kronecker quiver of dimension vector $(2,3)$, which is a blow down of the Hilbert scheme of 3 points on $\mathds{P}^2$. A short description of its geometry and Chow ring is given. Then we exhibit an exceptional sequence for the derived category by understanding a $\mathds{P}^1$-bundle over it and using Teleman Quantization. The fullness of the exceptional sequence is proved by using a covering argument and computations of mutations.