In this paper we extend the generalized algebraic fundamental group constructed in [EH] to general fibered categories using the language of gerbes. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined in [BV] for non smooth non pseudo-proper algebraic stacks.
We construct the moduli stack of torsors over the formal punctured disk in characteristic
p
>
0
p>0
for a finite group isomorphic to the semidirect product of a
p
p
-group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.
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. 1 Introduction Let G be a flat, finite group scheme finitely presented over a base scheme (say over a field, or, as in this paper, over Z). In this paper, we study G-Galois covers of very general schemes. We define a (ramified) G-cover as a finite morphism f : X −→ Y with an action
We study the stack B h,g,n of uniform cyclic covers of degree n between smooth curves of genus h and g and, for h ≫ g, present it as an open substack of a vector bundle over the universal Jacobian stack of Mg. We use this description to compute the integral Picard group of B h,g,n , showing that it is generated by tautological classes of B h,g,n .
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