Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-11T15:44:14.592Z Has data issue: false hasContentIssue false
  • English
  • Français

Deformations of elliptic fiber bundles in positive characteristic

Published online by Cambridge University Press:  11 January 2016

Holger Partsch
Holger Partsch*
Affiliation:
Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany, partsch@math.uni-duesseldorf.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

Keywords

14J2714D15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 211 , September 2013 , pp. 79 - 108
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[ABD+] Artin, M., Bertin, J. E., Demazure, M., Gabriel, P., Grothendieck, A., Raynaud, M., and Serre, J.-P., Schémas en groupes, Fasc. 7: Exposés 23 à 26, Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques 1963/64 (SGA 3), 1st ed., Inst. Hautes Etudes Sci., Paris, 1965/1966. MR 0207710.Google Scholar
[B] B˘adescu, L., Algebraic Surfaces, Universitext, Springer, New York, 2001. MR 1805816.Google Scholar
[BS] Basile, C. L. and Skorobogatov, A., “On the Hasse principle for bielliptic surfaces” in Number Theory and Algebraic Geometry, London Math. Soc. Lecture Note Ser. 303, Cambridge University Press, Cambridge, 2003, 31–40. MR 2053453.Google Scholar
[BM] Bombieri, E. and Mumford, D., “Enriques’ classification of surfaces in char. p, II” in Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 23–42. MR 0491719.CrossRefGoogle Scholar
[GM] Green, B. and Matignon, M., Liftings of Galois covers of smooth curves, Compos. Math. 113 (1998), 237–272. MR 1645000. DOI 10.1023/A:1000455506835.CrossRefGoogle Scholar
[Gr1] Grothendieck, A., Éléments de géométrie algébrique, II: Etude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Etudes Sci. 8, 1961. MR 0217084.Google Scholar
[Gr2] Grothendieck, A., Revätements étales et groupe fondamental, Fasc. II: Exposés 6, 8 à 11, Séminaire de Géométrie Algébrique 1960/1961, Inst. Hautes Etudes Sci., Paris, 1963. MR 0217088.Google Scholar
[Gr3] Grothendieck, A., Groupes de Barsotti-Tate et cristaux de Dieudonné, Sémin. Math. Super. 45, Université de Montréal, Montreal, 1974. MR 0417192.Google Scholar
[I] Illusie, L., “Grothendieck’s existence theorem in formal geometry” in Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, 2005, 179–233. MR 2223409.Google Scholar
[JLR] Jarvis, T. J., Lang, W. E., and Ricks, J. R., Integral models of extremal rational elliptic surfaces, Comm. Algebra 40 (2012), 3867–3883.CrossRefGoogle Scholar
[KU] Katsura, T. and Ueno, K., On elliptic surfaces in characteristic p, Math. Ann. 272 (1985), 291–330. MR 0799664. DOI 10.1007/BF01455561.CrossRefGoogle Scholar
[K] Katz, N. M., “Serre-Tate local moduli” in Algebraic Surfaces (Orsay, 1976–1978), Lecture Notes in Math. 868, Springer, Berlin, 1981, 138–202. MR 0638600.Google Scholar
[KM] Katz, N. M. and Mazur, B., Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud. 108, Princeton University Press, Princeton, 1985. MR 0772569.Google Scholar
[Kl] Kleiman, S. L., “The Picard scheme” in Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, 2005, 235–321. MR 2223410.Google Scholar
[L] Lang, W. E., Examples of liftings of surfaces and a problem in de Rham cohomology, Compos. Math. 97 (1995), 157–160. MR 1355122.Google Scholar
[LLR] Liu, Q., Lorenzini, D., and Raynaud, M., Néron models, Lie algebras, and reduction of curves of genus one, Invent. Math. 157 (2004), 455–518. MR 2092767. DOI 10. 1007/s00222-004-0342-y.CrossRefGoogle Scholar
[M] Milne, J. S., Etale Cohomology, Princeton Math. Ser. 33, Princeton University Press, Princeton, 1980. MR 0559531.Google Scholar
[Mu] Mumford, D., Abelian Varieties, Tata Inst. Fund. Res. Stud. Math. 5, Oxford University Press, London, 1970. MR 0282985.Google Scholar
[MFK] Mumford, D., Fogarty, J., and Kirwan, F., Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994. MR 1304906. DOI 10. 1007/978-3-642-57916-5.Google Scholar
[PS] Pacheco, A. and Stevenson, K. F., Finite quotients of the algebraic fundamental group of projective curves in positive characteristic, Pacific J. Math. 192 (2000), 143–158. MR 1741024. DOI 10.2140/pjm.2000.192.143.CrossRefGoogle Scholar
[R] Raynaud, M., Faisceaux amples sur les schémas en groupes et les espaces homogénes, Lecture Notes in Math. 119, Springer, Berlin, 1970. MR 0260758.Google Scholar
[S] Schlessinger, M., Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MR 0217093.CrossRefGoogle Scholar
[Si] Silverman, J. H., The Arithmetic of Elliptic Curves, 2nd ed., Grad. Texts in Math. 106, Springer, Dordrecht, 2009. MR 2514094. DOI 10.1007/978-0-387-09494-6.Google Scholar