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Deformation of torsors under monogenic group schemes. (English. French summary) Zbl 1346.14064
The following theorem is proved: Assume that \(\mathrm{Lie }G_k\) is of dimension \(\leq 1\) and that \(Y_k\) does not arise as the push-forward of a torsor over \(X_k\) under a proper subgroup scheme of \(G_k\). Then, there exist a smooth formal curve \(X\) over \(R\) and a \(G\)-torsor \(Y \to X\) whose special fiber is the \(G_k\)-torsor \(Y_k \to X_k\).
Here \(R\) is a complete local ring with residue field \(k\) of positive characteristic \(p >0\), \(G\) is a finite, flat and of finite presentation, commutative group scheme over \(R\) and \(X_k\) is a smooth curve over \(k\).
This article extends some earlier works of the authors [J. Algebra 318, No. 2, 1057–1067 (2007; Zbl 1135.14036)].
In the acknowledgements, the authors indicate that they ‘would like to thank M. Raynaud who suggested the problem to us and for his encouragement.’
The interesting feature of the article is the use of the equivariant cotangent complex by L. Illusie [Complexe cotangent et déformations. II. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0238.13017)], results on algebraic spaces and schemes by M. Raynaud and L. Gruson [Invent. Math. 13, 1–89 (1971; Zbl 0227.14010)], the analogue of the stack by D. Abramovich et al. [J. Algebr. Geom. 20, No. 3, 399–477 (2011; Zbl 1225.14020); corrigendum ibid. 24, No. 2, 399–400 (2015)] and moduli of Galois \(p\)-covers by D. Abramovich and M. Romagny [Algebra Number Theory 6, No. 4, 757–780 (2012; Zbl 1271.14032)].
In the last section of the paper under review, the authors show that these techniques can also be applied to the theory moduli of \(p\)-covering of curves. Several interesting examples are given, in particular on relation with Jacobians.
14G20 Local ground fields in algebraic geometry
11G20 Curves over finite and local fields
14H30 Coverings of curves, fundamental group
14C15 (Equivariant) Chow groups and rings; motives
Full Text: DOI
[1] D. Abramovich, M. Olsson & A. Vistoli, « Twisted stable maps to tame Artin stacks », J. Algebraic Geom.20 (2011), no. 3, p. 399-477. · Zbl 1225.14020
[2] D. Abramovich & M. Romagny, « Moduli of Galois \(p\)-covers in mixed characteristics », Algebra Number Theory6 (2012), no. 4, p. 757-780. · Zbl 1271.14032
[3] F. Andreatta & C. Gasbarri, « Torsors under some group schemes of order \(p^n\) », J. Algebra318 (2007), no. 2, p. 1057-1067. · Zbl 1135.14036
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[9] F. Oort & J. Steenbrink, « The local Torelli problem for algebraic curves », in Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, p. 157-204.
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