Stacks of ramified abelian covers

TL;DR

This paper introduces and studies the stack of ramified Galois covers for finite diagonalizable group schemes, revealing its structure, connectedness, and relation to equivariant Hilbert schemes, with a focus on the abelian case.

Contribution

It generalizes G-torsors to ramified Galois covers, analyzes the structure of the stack G-Cov, and establishes parametrization and toric descriptions in the abelian case.

Findings

G-Cov is connected for finite diagonalizable G over Z.

Contains a special irreducible component Z_G related to BG.

Provides conditions for constructing G-covers via parametrization maps.

Abstract

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-Cov, mainly in the abelian case, precisely when G is a finite diagonalizable group scheme over Z. In this case we prove that G-Cov is connected, but it is irreducible or smooth only in few finitely many cases. On the other hand, it contains a 'special' irreducible component Z_G, which is the closure of BG and this reflects the deep connection we establish between G-Cov and the equivariant Hilbert schemes. We introduce 'parametrization' maps from smooth stacks, whose objects are collections of invertible sheaves with additional data, to 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 Z_G is provided.