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Unformatted text preview: INTEGRAL MODELS FOR MODULI SPACES OF G-TORSORS MARTIN OLSSON 1. Introduction The work in this paper is a generalizations to higher dimensions of a particular application of the Abramovich-Vistoli theory of twisted stable maps [AV] (and its ‘tame version’ of Abramovich-Olsson-Vistoli [AVO2]). Let us begin by reviewing how the Abramovich-Vistoli theory gives compactifications of moduli spaces for curves with (possibly non-abelian) level structure. 1.1. Let C/S be a smooth proper curve of genus g over a scheme S , and let P be a finite set of prime numbers which includes all residue characteristics of S . For any section s : S → C we then obtain, as in [DM, 5.5], a pro-object π 1 ( X/S,s ) in the category of locally constant sheaves of finite groups on S whose fiber over a geometric ¯ t → S is equal to the maximal prime to P quotient of π 1 ( C ¯ t ,s ¯ t ). Now let G be a finite group of order not in P . Let H om ext ( π 1 ( X/S,s ) ,G ) denote the sheaf of homomorphisms π 1 ( X/S,s ) → G modulo the action of π 1 ( X/S,s ) given by conjugation. Then the sheaf H om ext ( π 1 ( X/S,s ) ,G ) is a locally constant sheaf on S which is canonically independent of the section s . It follows that for any smooth proper curve C/S of genus g there is a canonically defined sheaf H om ext ( π 1 ( X/S ) ,G ) even when C/S does not admit a section. Following [DM, 5.6], we define a Teichm¨uller structure of level G on C/S to be a section of H om ext ( π 1 ( X/S ) ,G ), which ´ etale locally on S can be represented by a surjective homomorphism π 1 ( X/S,s ) → G for a suitable section s . As in [DM, 5.8] we define G M g to be the stack over Z [1 / | G | ] which to any Z [1 / | G | ]-scheme S associates the groupoid of pairs ( C/S,σ ), where C/S is a smooth proper genus g curve and σ is a Teichm¨uller structure of level G . 1.2. The space G M g is connected with the Abramovich-Vistoli theory as follows. Let C/S be a curve as above, and fix a section s : S → C . Let G K ◦ g denote the stack over Z [1 / | G | ] which to any Z [1 / | G | ]-scheme S associates the groupoid of principal G-bundles P → C , such that for every geometric point ¯ t → S the fiber P ¯ t → C ¯ t is connected. The choice of the section s enables us to describe the stack G K ◦ g as follows. For any object P → C of G K ◦ g , the pullback s * P is a G-torsor with action of π 1 ( X/S,s ) on S . ´ Etale locally on S we can choose a trivialization ˜ s : S → s * P of the G-torsor s * P , and such a trivialization defines a homomorphism π 1 ( X/S,s ) → G. The assumption on the connectedness of the geometric fibers P ¯ t implies that this map is surjective. It follows that the conjugacy class of the homomorphism π 1 ( X/S,s ) → G is independent of the choice of ˜ s and also independent of the section s . In this way we obtain 1 2 MARTIN OLSSON a morphism (1.2.1) G K ◦ g → G M g ....
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This note was uploaded on 02/05/2011 for the course MATH 224b taught by Professor Grunbaum,f during the Spring '08 term at Berkeley.
- Spring '08