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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)

MSC:

14H30 Coverings of curves, fundamental group
14L15 Group schemes
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References:

[1] Anantharaman S.: Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1. Mémoires de la S. M. F., tome 33, 5–79 (1973) · Zbl 0286.14001
[2] Antei M.: Comparison between the fundamental group scheme of a relative scheme and that of its generic fibre. J. théorie des nombres de Bordeaux, Tome 22(3), 537–555 (2010) · Zbl 1267.14029 · doi:10.5802/jtnb.731
[3] Antei M.: On the Abelian fundamental group scheme of a family of varieties. Israel J. Math. 186(1), 427–446 (2011) · Zbl 1263.14047 · doi:10.1007/s11856-011-0147-9
[4] Antei M.: The fundamental group scheme of a non reduced scheme. Bull. Sci. Math. 135(5), 531–539 (2011) · Zbl 1223.14054 · doi:10.1016/j.bulsci.2011.02.002
[5] Artin, M.: Lipman’s Proof of Resolution of Singularities for Surfaces, on Arithmetic Geometry, pp. 267–288. Springer, New York (1995)
[6] Bertin, J.E.: Généralites sur les préschémas en groupes. Éxposé VI B , Séminaires de géométrie algébrique du Bois Marie. III (1962/1964)
[7] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Springer, New York (1980)
[8] Demazure M., Gabriel P.: Groupes algébriques. North-Holland Publ. Co., Amsterdam (1970) · Zbl 0203.23401
[9] Garuti M.A.: On the ”Galois closure” for torsors. Proc. Am. Math. Soc. 137, 3575–3583 (2009) · Zbl 1181.14053 · doi:10.1090/S0002-9939-09-09997-3
[10] Grothendieck, A.: Éléments de Géométrie Algébrique. I. Le langage des schémas. Publications Mathématiques de l’IHES 4, (1960)
[11] Grothendieck, A.: Revêtements étales et groupe fondamental, Séminaire de géométrie algébrique du Bois Marie (1960–1961)
[12] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics. Springer, New York (1977) · Zbl 0367.14001
[13] Kleiman, S.L.: The Picard Scheme. Fundamental Algebraic Geometry. AMS, Providence (2005) · Zbl 1105.14060
[14] Lipman J.: Desingularization of two-dimensional schemes. Ann. Math. 107, 151–207 (1978) · doi:10.2307/1971141
[15] Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford Science Publications, Oxford (2002) · Zbl 0996.14005
[16] Maugeais S.: Relèvement des revêtements p-cycliques des courbes rationnelles semistables. Math. Ann. 327(2), 365–393 (2003) · Zbl 1073.14042 · doi:10.1007/s00208-003-0458-1
[17] Nori M.V.: The fundamental group scheme of an Abelian variety. Math. Ann. 263, 263–266 (1983) · doi:10.1007/BF01457128
[18] Oort F., Sekiguchi T., Suwa N.: On the deformation of Artin-Schreier to Kummer. Ann. Sci. Éc. Norm. Sup. (4-ème série) 22(3), 345–375 (1989) · Zbl 0714.14024
[19] Raynaud, M.: p-groupes et réduction semi-stable des courbes. The Grothendieck Festschrift, vol. III, Progr. Math., vol. 88, pp. 179–197. Birkhäuser, Boston (1990)
[20] Saïdi M.: Revêtements Étales Abéliens Courants sur les Graphes et Réduction Semi-Stable des courbes. Manuscr. Math. 89(2), 245–265 (1996) · Zbl 0869.14010 · doi:10.1007/BF02567516
[21] Saïdi M.: Torsors under finite and flat group schemes of rank p with Galois action. Math. Zeit. 245(4), 695–710 (2003) · Zbl 1055.14048 · doi:10.1007/s00209-003-0566-3
[22] Shatz, S.S.: Group Schemes, Formal Groups, and p-Divisible Groups, on Arithmetic Geometry, pp. 29–78. Springer, New York (1995)
[23] Szamuely, T.: Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, vol. 117. Cambridge University Press, Cambridge (2009) · Zbl 1189.14002
[24] Waterhouse W.C., Weisfeiler B.: One-dimensional affine group schemes. J. Algebra 66, 550–568 (1980) · Zbl 0452.14013 · doi:10.1016/0021-8693(80)90104-0
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