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Let $\rho :\mathbb{Z}/k \mathbb{Z}\rightarrow Sl(2,\mathbb{C})$ be arepresentation of a finite abelian group and let $\Theta^{gen}\subsetHom_\mathbb{Z}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of genericstability conditions on the set of $G$-constellations. We provide acombinatorial description of all the chambers $C\subset\Theta^{gen}$ and provethat there are $k!$ of them. Moreover, we introduce the notion of simplechamber and we show that, in order to know all toric $G$-constellations, it isenough to build all simple chambers. We also prove that there are $k\cdot2^{k-2} $ simple chambers. Finally, we provide an explicit formula for thetautological bundles $\mathscr{R}_C$ over the moduli spaces $\mathscr{M}_C$ forall chambers $C\subset \Theta^{gen}$ which will only depend upon acombinatorial object, called chamber stair, attached to the chamber $C$.

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