Stacks of ramified Galois covers

Abstract

Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the category of finite locally free sheaves over $T$, extending the classical Tannakian correspondence between $G$-torsors and strong monoidal functors as above. Using this point of view, I prove that $G$-Cov is always reducible if $G$ is a non-abelian linearly reductive group. When $G$ is constant and tame I also give a criterion to detect when a $G$-cover of a regular in codimension one, integral scheme has regular in codimension one total space in terms of the functor associated with the cover. In the last part I focus on the case $G=S_3$, prove that $S_3$-Cov has exactly two irreducible components and describe the principal one. I also describe particular open loci of $S_3$-Cov, that is particular families of $S_3$-covers, classify $S_3$-covers of regular schemes whose total space is regular and compute the invariants of $S_3$-covers of smooth surfaces.