Moduli spaces of $\mathbb{Z}/k\mathbb{Z}$-constellations over $\mathbb{A}^2$

TL;DR

This paper characterizes the structure of moduli spaces of $ ext{Z}/k ext{Z}$-constellations over the affine plane, revealing the combinatorial nature of stability chambers and providing explicit formulas for associated tautological bundles.

Contribution

It offers a complete combinatorial description of stability chambers, introduces the concept of simple chambers, and derives explicit formulas for tautological bundles over the moduli spaces.

Findings

Number of chambers is $k!$.

Number of simple chambers is $k imes 2^{k-2}$.

Explicit formulas for tautological bundles depending on chamber stairs.

Abstract

Let $\rho:\mathbb{Z}/k \mathbb{Z}\rightarrow \text{SL}(2,\mathbb{C})$ be a representation of a finite abelian group and let $\Theta^{\text{gen}}\subset \text{Hom}_\mathbb{Z}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of generic stability conditions on the set of $G$-constellations. We provide a combinatorial description of all the chambers $C\subset\Theta^{\text{gen}}$ and prove that there are $k!$ of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric $G$-constellations, it is enough to build all simple chambers. We also prove that there are $k\cdot 2^{k-2} $ simple chambers. Finally, we provide an explicit formula for the tautological bundles $\mathscr{R}_C$ over the moduli spaces $\mathscr{M} _C$ for all chambers $C\subset \Theta^{\text{gen}}$ which only depends upon the chamber stair which is a combinatorial object attached to the chamber $C$.