Extension of finite solvable torsors over a curve. (English) Zbl 1276.14043

Suppose that \(S\) is a connected Dedekine scheme with \(\mathrm{dim}(S)=1\) and let \(\eta = \mathrm{Spec}(K)\) be its generic point. Let \(X\) be \(S\)-scheme that is faithfully flat of finite type. Let \(G\) be a finite \(K\)-group scheme and \(Y \to X_{\eta}\) a \(G\)-torsor. To extend the \(G\)-torsor \(Y \to X_{\eta}\) to \(X\) involves finding a finite flat \(S\)-group scheme \(G'\) with generic fibre \(G\), and a \(G'\)-torsor \(T \to X\) with generic fibre isomorphic to \(Y \to X_{\eta}\). A. Grothendieck [Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag. XXII, 447 p. (1971; Zbl 0234.14002)], M. Raynaud [in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 179–197 (1990; Zbl 0722.14013)], M. Saïdi [Manuscr. Math. 89, No. 2, 245–265 (1996; Zbl 0869.14010)] and the author have provided partial solutions before. In this paper, the author provides another possible solution to this problem as follows (Theorem 1.1):
Theorem. Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and with algebraically closed residue field of characteristic \(p>0\). Let \(X\) be a smooth fibered surface over \(R\). Let \(G\) be a finite, étale and solvable \(K\)-group scheme. Then for every connected and pointed \(G\)-torsor \(Y\) over the generic fibre \(X_{\eta}\) of \(X\) there exists a regular fibered surface \(\tilde{X}\) over \(R\) and a model map \(\tilde{X} \to X\) such that \(Y\) can be extended to a torsor over \(\tilde{X}\) possibly after extending scalars in the following two cases: 1) \(|G| = p^n\); 2) \(G\) has a normal series of length \(2\).
To prove the theorem, first decompose \(Y \to X_{\eta}\) into a tower of quotient pointed \((\mathbb{Z}/p\mathbb{Z})_K\)-torsors after extending the scalars. The author (Theorem 3.12) has proved that when the Jacobian has abelian reduction, then the torsor can be extended to \(X\). If necessary, desingularize so that the torsor is regular (Theorem 3.8). Then proceed with induction.
Reviewer: Xiao Xiao (Utica)


14H30 Coverings of curves, fundamental group
14L15 Group schemes
Full Text: DOI arXiv


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