no. 4
Formal deformation of curves with group scheme action
Annales de l'Institut Fourier, Volume 55 (2005) no. 4, pp. 1105-1165.

We study equivariant deformations of singular curves with an action of a finite flat group scheme, using a simplified version of Illusie’s equivariant cotangent complex. We apply these methods in a special case which is relevant for the study of the stable reduction of three point covers.

Nous étudions les déformations équivariantes d’une courbe singulière sur laquelle agit un schéma en groupes, en utilisant une version simplifiée du complexe cotangent équivariant définie par Illusie. Ces méthodes sont explicitées dans un cas particulier qui apparaît naturellement dans l’étude des revêtements galoisiens étale de la droite projective moins trois points.

DOI: 10.5802/aif.2122
Classification: 14B12,  14H30,  13D10
Keywords: Equivariant deformation, curves, group schemes, cotangent complex 
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Wewers, Stefan. Formal deformation of curves with group scheme action. Annales de l'Institut Fourier, Volume 55 (2005) no. 4, pp. 1105-1165. doi : 10.5802/aif.2122. https://aif.centre-mersenne.org/articles/10.5802/aif.2122/

[1] M. Artin; A. Grothendieck; J.L. Verdier Théorie des Topos et Cohomologie Étale des Schémas (SGA 4), Lecture Notes in Math., Tome 269, 270, 305 (1973), p. Springer-Verlag | MR: 354654

[2] P. Berthelot; A. Grothendieck; L. Illusie Théorie des intersections et théorème de Riemann-Roch (SGA 6), Lecture Notes in Math., Tome 225, Springer-Verlag, 1971 | MR: 354655 | Zbl: 0218.14001

[3] J. Bertin; A. Mézard Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math., Tome 141 (2000), pp. 195-238 | Article | MR: 1767273 | Zbl: 0993.14014

[4] P. Deligne; D. Mumford The irreducibility of the space of curves of given genus, Publ. Math. IHES, Tome 36 (1969), pp. 75-109 | Numdam | MR: 262240 | Zbl: 0181.48803

[5] J. Giraud Cohomologie non abélienne, Grundlehren, Tome 179, Springer-Verlag, 1971 | MR: 344253 | Zbl: 0226.14011

[6] B. Green; M. Matignon Lifting of Galois covers of smooth curves, Compositio Math., Tome 113 (1998) no. 3, pp. 237-272 | Article | MR: 1645000 | Zbl: 0923.14006

[7] A. Grothendieck Sur quelques points d'algèbre homologique, Tôhoku Math. J., Tome 9 (1957), pp. 119-221 | MR: 102537 | Zbl: 0118.26104

[8] A. Grothendieck Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Math., Tome 79 (1968) no. Springer-Verlag | MR: 241495 | Zbl: 0201.53803

[9] A. Grothendieck; M. Demazure Schémas en Groupes I, II, III (SGA 3), Lecture Notes in Math., Tome 151, 152, 153, Springer-Verlag, 1970 | MR: 274458 | Zbl: 0207.51401

[10] R. Hartshorne Algebraic Geometry, GTM, Tome 52, Springer-Verlag, 1977 | MR: 463157 | Zbl: 0367.14001

[11] Y. Henrio Arbres de Hurwitz et automorphismes d'ordre $p$ des disques et des couronnes $p$-adiques formels (math.AG/0011098, http://arxiv.org/abs/math.AG/0011098)

[12] L. Illusie Complexe cotangent et déformations I+II, Lecture Notes in Math., Tome 239, 283, Springer-Verlag, 1971, 1972 | MR: 491680 | Zbl: 0224.13014

[13] L. Illusie; F.W. Lawvere Cotangent complex and deformations of torsors and group schemes, Toposes, Algebraic Geometry and Logic, Tome 274 (1972), pp. 159-189 | Zbl: 0238.14016

[14] G. Laumon; L. Moret-Bailly Champs Algébriques, Ergebnisse der Math. und ihrer Grenzgebiete, Tome 39, Springer--Verlag, 2000 | MR: 1771927 | Zbl: 0945.14005

[15] H. Matsumura Commutative Algebra, Benjamin, 1980 | MR: 575344 | Zbl: 0441.13001

[16] H. Matsumura Commutative Ring Theory, Cambridge studies in adv. math., Tome 8, Cambridge Univ. Press, 1986 | MR: 879273 | Zbl: 0603.13001

[17] D. Mumford Abelian Varieties, Oxford Univ. Press, 1970 | MR: 282985 | Zbl: 0326.14012

[18] M. Raynaud Spécialisation des revêtements en caractéristique $p>0$, Ann. Sci. École Norm. Sup., Tome 32 (1999) no. 1, pp. 87-126 | Numdam | MR: 1670532 | Zbl: 0999.14004

[19] M. Saïdi Torsors under finite and flat group schemes of rank $p$ with Galois action (math.AG/0106246, http://arxiv.org/abs/math.AG/0106246)

[20] M. Schlessinger Functors of Artin rings, Trans. Am. Math. Soc., Tome 130 (1968), pp. 208-222 | Article | MR: 217093 | Zbl: 0167.49503

[21] T. Sekiguchi; F. Oort; N. Suwa On the deformation of Artin--Schreier to Kummer, Ann. Sci. École Norm. Sup., Tome 22 (1989), pp. 345-375 | Numdam | MR: 1011987 | Zbl: 0714.14024

[22] K.H. Ulbrich Group cohomology for Picard categories, J. Algebra, Tome 91 (1984) | MR: 769585 | Zbl: 0554.18005

[23] A. Vistoli The deformation theory of local complete intersections (math. AG/9703008, http://arxiv.org/abs/alg-geom/9703008)

[24] C.A. Weibel An introduction to homological algebra, Cambridge studies in advanced mathematics, Tome 38, Cambridge Univ. Press, 1994 | MR: 1269324 | Zbl: 0797.18001

[25] S. Wewers Reduction and lifting of special metacyclic covers, Ann. Sci. École Norm. Sup., Tome 36 (2003), pp. 113-138 | Numdam | MR: 1987978 | Zbl: 1042.14005

[26] S. Wewers Three point covers with bad reduction, J. Amer. Math. Soc., Tome 16 (2003) no. 4, pp. 991-1032 | Article | MR: 1992833 | Zbl: 1062.14038

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