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

Abstract

Let $\ddot{I}:\mathbb{Z}/k\mathbb{Z}\to \text{SL}(2,\mathbb{C})$ be a representation of a finite abelian group and let $\hat{I}{˜}^{\text{gen}}\subset {\text{Hom}}_{\mathbb{Z}}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of generic stability conditions on the set of $G$-constellations. We provide a combinatorial description of all the chambers $C\subset \hat{I}{˜}^{\text{gen}}$ 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 ${\mathcal{R}}_C$ over the moduli spaces ${\mathcal{M}}_C$ for all chambers $C\subset \hat{I}{˜}^{\text{gen}}$ which only depends upon the chamber stair which is a combinatorial object attached to the chamber $C$.

44 pages, 21 figures