Abstract
Let Y K → X K be a Galois covering of smooth curves over a field of characteristic 0, with Galois group G. We assume K is the fraction field of a discrete valuation ring R with residue characteristic p. Assuming p 2 ∤ G and the p-Sylow subgroup of G is normal, we consider the possible reductions of the covering modulo p. In our main theorem we show the existence, after base change, of a twisted curve \(\mathcal{X} \rightarrow Spec (R)\), a group scheme \(\mathcal{G}\rightarrow \mathcal{X}\) and a covering \(Y \rightarrow \mathcal{X}\) extending Y K → X K , with Y a stable curve, such that Y is a \(\mathcal{G}\)-torsor.In case p 2 | G counterexamples to the analogous statement are given; in the appendix a strong counterexample is given, where a non-free effective action of α p 2 on a smooth 1-dimensional formal group is shown to lift to characteristic 0.
In grateful memory of Serge Lang
Mathematics Subject Classification (2010): 14H25, 14H30
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D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), 3547–3618.
D. Abramovich, T. Graber and A. Vistoli Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.
D. Abramovich, M. Olsson and A. Vistoli, Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 4, 1057–1091.
D. Abramovich, M. Olsson and A. Vistoli, Twisted stable maps to tame Artin stacks, J. Alg. Geom S 1056–3911 (2010) 00569-3; published electronically.
D. Abramovich and F. Oort, Stable maps and Hurwitz schemes in mixed characteristic. Advances in algebraic geometry motivated by physics (Lowell, MA, 2000), E. Previato, ed., 89–100, Contemp. Math., 276, Amer. Math. Soc., Providence, RI, 2001.
D. Abramovich and A. Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75.
D. Abramovich and A. Vistoli, Twisted stable maps and quantum cohomology of stacks, Intersection theory and moduli, 97–138, ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.
I. I. Bouw, Reduction of the Hurwitz space of metacyclic covers, Duke Math. J. 121 (2004), no. 1, 75–111.
I. I. Bouw and S. Wewers, Reduction of covers and Hurwitz spaces, J. Reine Angew. Math. 574 (2004), 1–49.
D. Abramovich and A. Vistoli, Stable reduction of modular curves, Mathematisches Institut, Georg-August-Universität Göttingen: Seminars 2003/2004, 37–41, Universitätsdrucke Göttingen, Gttingen, 2004.
T. Ekedahl, Boundary behaviour of Hurwitz schemes. The moduli space of curves (Texel Island, 1994), 173–198, Prog. Math., 129, Birkhäuser Boston, Boston, MA, 1995.
Y. Henrio, Arbres de Hurwitz et automorphismes d’ordre p des disques et des couronnes p-adiques formels, preprint math.AG/0011098.
J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88.
J. Lubin and J. Tate, Formal complex multiplication in local fields, Annals of Math. 81 (1965), 380–387.
S. Maugeais, On a compactification of a Hurwitz space in the wild case, preprint math.AG/0509118.
S. Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 355–441.
M. Olsson, On (log) twisted curves, Compos. Math. 143 (2007), no. 2, 476–494.
F. Oort and J. Tate, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4), 3 (1970), 1–21.
M. Raynaud, Spécialisation des revêtements en caractéristique p > 0. Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 87–126.
M. Romagny, Sur quelques aspects des champs de revêtements de courbes algébriques, Ph.D. Thesis, University of Grenoble, 2002.
M. Romagny-, Effective models of group schemes, preprint arXiv:0904.3167.
M. Saidi, Torsors under finite flat group schemes of rank p with Galois action, Math. Z. 245 (2003), no. 4, 695–710.
M. Saidi-, Wild ramification and a vanishing cycle formula, J. Algebra 273 (2004), no. 1, 108–128.
M. Saidi--, Galois covers of degree p and semi-stable reduction of curves in mixed characteristics. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, 661–684.
M. Saidi---, On the existence of a torsor structure for Galois covers, preprint math.AG/0403389
Dajano Tossici, Models of \(\mathbb{Z}/{p}^{2}\mathbb{Z}\) over a d.v.r. of unequal characteristic, Models of \({\mu }_{{p}^{2},k}\) over a discrete valuation sing; with appendix by Xavier Caruso. J. Aligebra 323 (2010), no. 7, 1908–1957.
Dajano Tossici, Effective models and extension of torsors over a discrete valuation ring of unequal characteristic, Int. Math. Res. Not. IMRN 2008, Art. ID rnn111, 68 pp.
Wolmer Vasconcelos, Integral closure. Rees algebras, multiplicities, algorithms. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.
S. Wewers, Construction of Hurwitz spaces, Institut für Experimentelle Mathematik preprint No. 21 (1998).
S. Wewers, Reduction and lifting of special metacyclic covers, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 1, 113–138.
S. Wewers-, Three point covers with bad reduction, J. Amer. Math. Soc. 16 (2003), no. 4, 991–1032.
S. Wewers--, Formal deformation of curves with group scheme action, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 4, 1105–1165.
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Abramovich, D., Lubin, J. (2012). Raynaud’s group-scheme and reduction of coverings. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_1
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