Stacks of Ramified Covers Under Diagonalizable Group Schemes

Given a flat, finite group scheme G finitely presented over a base scheme we introduce the notion of ramified Galois cover of group G (or simply G-cover), which generalizes the notion of G-torsor. We study the stack of G-covers, denoted with G, mainly in the abelian case, precisely when G is a finite diagonalizable group scheme over $\mathbb Z$. In this case, we prove that G is connected, but it is irreducible or smooth only in few finitely many cases. On the other hand, it contains a “special” irreducible component $\mathcal Z_G$, which is the closure of G and this reflects the deep connection we establish between G and the equivariant Hilbert schemes. We introduce “parametrization” maps from smooth stacks, whose objects are collections of invertible sheaves with additional data, to $\mathcal Z_G$ and we establish sufficient conditions for a G-cover in order to be obtained (uniquely) through those constructions. Moreover, a toric description of the smooth locus of $\mathcal Z_G$ is provided.