Integral models for moduli spaces of \(G\)-torsors
This article makes a generalization of an application of the Abramovich-Vistoli theory of twisted stable maps to higher dimensions. It is a general technique to use the Abromovich-Vistoli theory to construct compactifications of moduli spaces of curves with level structure. For the following, let \(g\) be an integer \(\geq 2\). Given \(C/S\) a smooth proper curve of genus \(g\) over a scheme \(S\). Let \(P\) be a finite set of primes which includes all residue characteristics of \(S\). For any section \(s:S\rightarrow C\) there exists a pro-object \(\pi_1(C/S,s)\) in the category of locally constant sheaves of finite groups on \(S\) whose fiber over a geometric \(\overline{t}\rightarrow S\) is equal to the maximal quotient of \(\pi_1(C_{\overline t},s_{\overline t})\) prime to \(P\). For a finite group \(G\) of order not divisible by any of the primes in \(P\), \(\mathcal H om^{\text{ext}}(\pi_1(C/S,s),G)\) denotes the sheaf of homomorphisms \(\pi_1(C/S,s)\rightarrow G\) modulo the action of \(\pi_1(C/S,s)\) given by conjugation. Then \(\mathcal H om^{\text{ext}}(\pi_1(C/S,s),G)\) is locally constant on \(S\) and is canonically independent of the section \(s\). It turns out that for any smooth curve \(C/S\) this sheaf can be canonically defined even when there does not exist a section. A Teichmüller structure of level \(G\) on \(C/S\) is defined as a section of \(\mathcal H om^{\text{ext}}(\pi_1(C/S),G)\) which étale locally on \(S\) can be represented by a surjective homomorphism \(\pi_1(C/S),s)\rightarrow G\) for a section \(s\). The stack \({}_G\mathcal M_g\) over \(\mathbb Z[1/|G|]\) is defined by the condition that to any \(\mathbb Z[1/|G|]\)-scheme \(S\) is associated the groupoid of pairs \((C/S,\sigma)\), where \(C\) is a smooth proper curve of genus \(g\) over \(S\) and \(\sigma\) is a Teichmüller structure of level \(G\) on \(C/S\). The connection with the theory of Abramovich-Vistoli is described in short form as follows: Let \({}_G\mathcal K_g^\circ\) be the \(\mathbb Z[1/|G|]\)-stack associating to a \(\mathbb Z[1/|G|]\)-scheme \(S\) the groupoid of pairs \((C/S,P\rightarrow C)\) with \(C/S\) a smooth proper genus \(g\)-curve, and \(P\rightarrow S\) a principal \(G\)-bundle, such that for every geometric point \(\overline t\rightarrow S\) the fibre \(P_{\overline t}\rightarrow C_{\overline t}\) is connected. Then there is a morphism of stacks \({}_G\mathcal K_g^\circ\rightarrow\mathcal M_g\) sending the pair \((C/S,P\rightarrow C)\) to the curve \(C/S\). Choose a trivialization \(\tilde s:S^\prime\rightarrow s^\ast P\) of the \(G\)-torsor \(s^\ast P\). It defines a homomorphism \(\pi_1(C/S,s)\rightarrow G\), and the conjugacy class of this homomorphism is independent of the choice of \(\tilde s\) and also of the section \(s\). Thus a section of \(\mathcal H om^{\text{ext}}(\pi_1(C/S),G)\) is obtained, even when \(C/S\) does not admit a section. This defines a morphism of stacks of \(\mathcal M_g\), \({}_G\mathcal K_g^\circ\rightarrow{}_G\mathcal M_g\) and this map identifies \({}_G\mathcal M_g\) with the relative coarse moduli space of \({}_G\mathcal K_g^\circ\rightarrow\mathcal M_g\). From another viewpoint, the category of \(G\)-torsors over a smooth genus \(g\)-curve \(C/S\) is equivalent to the category of morphisms \(C\rightarrow BG\). So the theory of twisted stable maps gives a natural compactification \({}_G\mathcal K_g\) of \({}_G\mathcal K_g^\circ\) over \(\mathbb Z[1/|G|]\). Forgetting the \(G\)-torsor gives a morphism \({}_G\mathcal K_g\rightarrow\overline{\mathcal M}_g\), so that passing to the associated relative coarse moduli space over \(\overline{\mathcal M}_g\) gives a compactification of \({}_G\mathcal M_g\). The author proves that if \(G\) is a tame group scheme it is possible to give a theory of twisted stable maps. This gives an extension of the stable map spaces to schemes where \(|G|\) is not invertible, and the resulting moduli spaces \({}_G\mathcal K_g\) are proper. Considering e.g. \(G=\mu_n\), the space \({}_{\mu_n}\mathcal K_g\) is a proper moduli space parameterizing \(\mu_n\)-torsors over twisted curves. The relative coarse moduli space of \({}_{\mu_n}\mathcal K_g^\circ\rightarrow\mathcal M_g\) is equal to the \(n\)-torsion subgroup scheme of the Jacobian of the universal curve over \(\mathcal M_g\). Thus proper models over \(\mathbb Z\) of the \(n\)-torsion subgroup of the universal Jacobian can be given. There are two natural parts in the construction of the compactification \({}_G\mathcal K_g\). There is the Deligne -Mumford compactification \(\mathcal M_g\hookrightarrow\overline{\mathcal M}_g\) of \(\mathcal M_g\), and there is the universal stable curve \(\mathcal C\rightarrow\overline{\mathcal M}_g\) restricting to the universal curve \(\mathcal C\rightarrow {\mathcal M}_g\). In the last case, viewing \(\mathcal M_g\) as the base, \({}_G\mathcal K_g\) is a compactification over \(\overline{\mathcal M}_g\) of the stack classifying \(G\)-torsors on a fixed family of curves \(\mathcal C\rightarrow\mathcal M_g\). The present article generalizes the second view to higher dimensional varieties. Consider a flat, proper, and semi-stable morphism of log schemes \(f:(X,M_X)\rightarrow (S,M_S)\). when \(G\) is a tame flat group scheme, the author constructs a proper \(S\)-stack \({}_G\mathcal K_{X/S}\) whose restriction to the open subset \(S^\circ\subset S\) where \(X\rightarrow S\) is smooth, is the moduli stack of \(G\)-torsors on \(X^\circ\) defined as \(X^\circ=X\times_S S^\circ\). If \(S=\overline{\mathcal M}_g\) and \(X=\overline{\mathcal C}\) is the universal curve over \(\overline{\mathcal M}_g\), the log structures \(M_{\overline{\mathcal M}_g}\) and \(M_{\overline{\mathcal C}}\) defined by the divisors at infinity define a morphism of log stacks \((\overline{\mathcal C},M_{\overline{\mathcal C}})\rightarrow (\overline{\mathcal M}_g,M_{\overline{\mathcal M}_g})\) and the present theory can be applied. In this way, the spaces of stable maps \({}_G\mathcal K_g\) is obtained. Correspondingly, for any simple extension of log structures \(j:M_S\hookrightarrow N_S\) on \(S\) there is an associated tame stack \(\mathcal X_j\rightarrow X\). Then \({}_G\mathcal K_{X/S}\) is the stack which to any \(S\) scheme \(g:T\rightarrow S\) associates the groupoid of pairs \((j:g^\ast M_S\hookrightarrow N_T,P\rightarrow\mathcal X_j),\) with \(j\) a simple extension with associated tame stack \(\mathcal X_j\rightarrow X\times_S T,\) and \(P\rightarrow\mathcal X_j\) a \(G\) torsor. Together with a suitable stability condition, this is called a \textit{twisted \(G\)-torsor}. The main property of the twisted \(G\)-torsor is a main result of the article: (i) The stack \({}_G\mathcal K_{X/S}\) is a proper algebraic stack over \(S\) with finite diagonal. (ii) The stack \({}_G\mathcal K_{X/S}\) is tame. To be able to prove the main results, the author studies various extension results for \(G\)-torsors with \(G\) a tame group scheme. Letting \(V\) be a discrete valuation ring, \(G/V\) a finite flat tame group scheme, and \(P_\eta\) a \(G\)-torsor over the field of fractions of \(V\). After a ramified base change of \(V\), the \(G\)-torsor \(P_\eta\) extends to \(V\). The problem of when a torsor on a stack descends to the coarse moduli space is studied, including the relative setting. Finally, it is proved that \({}_G\mathcal K_{X/S}\) is an Artin stack of finite type over the base \(S\) with finite diagonal. The article is sufficiently self contained and detailed. It is lot of parts to be proved to obtain the generalizations, and the author follows the procedure by proving point by point the different parts in a good way. Also, the article is a nice text for getting into the language of log schemes, and in particular stacks, at least if one consult the long list of relevant references.
scientific article; zbMATH DE number 6101193
scientific article