Stacks of Ramified Covers Under Diagonalizable Group Schemes

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. 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

Ramified Galois Covers via Monoidal Functors

“…When G is a diagonalizable group the same result holds except for a few cases when G has low rank (see [Ton13a,Corollary 4.17]). Thus the bad behaviour of the moduli G-Cov is still present in the nonabelian setting.…”

“…The stack G-Cov has been introduced in [Ton13a], it is algebraic and of finite type over R and contains B R G as an open substack.…”

“…Let R be a base commutative ring and G be a flat, finite and finitely presented group scheme over R. In [Ton13a] I introduced the notion of a ramified Galois cover with group G, briefly a G-cover, and the stack G-Cov of such objects (see 1.2 for details). This stack is algebraic and of finite type over R and contains B R G, the stack of G-torsors, as an open substack.…”

“…This stack is algebraic and of finite type over R and contains B R G, the stack of G-torsors, as an open substack. If G is diagonalizable, its nice representation theory makes it possible to study G-covers in terms of simplified data (collections of invertible sheaves and morphisms between them) and to investigate the geometry of the moduli G-Cov (see [Ton13a]). …”

Moduli of formal torsors

“…Notice that points of ∆ G over a eld L, namely G-torsors over L((t)), can also be seen as (not necessarily connected) Galois extensions of L((t)) and, taking integers, as special covers of L[[t]] with an action of G. It is therefore natural to ask and indeed this has been our initial approach to the problem, if one can dene a moduli space of special G-covers of B[[t]] for varying B or, more precisely, give a dierent moduli interpretation of G-torsors of B((t)) in terms of covers of B[[t]], in the spirit of [Ton17] and [Ton14]. We don't have a precise answer to this question, but in [TY19,Yas] we give partial answers.…”