Deformations of elliptic fiber bundles in positive characteristic. (English) Zbl 1274.14044

The paper under review concerns a question raised by Katsura and Ueno: can every elliptic surface over a field of positive characteristic be lifted to characteristic zero? Previously, non-liftable surfaces were only known to occur among quasi-elliptic surfaces and surfaces of general type. The author proves a negative answer for characteristics 2 and 3. More precisely, he proves that there are non-liftable elliptic fibre bundles, i.e., elliptic fibrations which are locally trivial in the étale topology.
The main ingredients of the construction include group actions on elliptic curves which are impossible in characteristic zero, and in particular, deformation theory. To this end, elliptic fiber bundles are considered as torsors over the relative Picard schemes. This allows the author to reduce to the Jacobian case and apply results from moduli theory for elliptic curves.
Applications include quotients of product surfaces \(E\times C\) and especially bielliptic surfaces which show several pathological phenomena such as obstructed deformations and deforming Jacobian fibrations into non-Jacobian fibrations.


14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14D15 Formal methods and deformations in algebraic geometry
Full Text: DOI arXiv Euclid


[1] M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, and J.-P. Serre, 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 Études Sci., Paris, 1965/1966.
[2] L. Bădescu, Algebraic Surfaces , Universitext, Springer, New York, 2001.
[3] C. L. Basile and A. Skorobogatov, “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. · Zbl 1073.14052
[4] E. Bombieri and D. Mumford, “Enriques’ classification of surfaces in char. \(p\), II” in Complex Analysis and Algebraic Geometry , Iwanami Shoten, Tokyo, 1977, 23-42. · Zbl 0348.14021
[5] B. Green and M. Matignon, Liftings of Galois covers of smooth curves , Compos. Math. 113 (1998), 237-272. · Zbl 0923.14006
[6] A. Grothendieck, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes , Publ. Math. Inst. Hautes Études Sci. 8 , 1961.
[7] A. Grothendieck, Revêtements étales et groupe fondamental, Fasc. II: Exposés 6, 8 à 11 , Séminaire de Géométrie Algébrique 1960/1961, Inst. Hautes Études Sci., Paris, 1963. · Zbl 0118.36206
[8] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné , Sémin. Math. Supér. 45 , Université de Montréal, Montreal, 1974. · Zbl 0331.14021
[9] L. Illusie, “Grothendieck’s existence theorem in formal geometry” in Fundamental Algebraic Geometry , Math. Surveys Monogr. 123 , Amer. Math. Soc., Providence, 2005, 179-233.
[10] T. J. Jarvis, W. E. Lang, and J. R. Ricks, Integral models of extremal rational elliptic surfaces , Comm. Algebra 40 (2012), 3867-3883. · Zbl 1260.14043
[11] T. Katsura and K. Ueno, On elliptic surfaces in characteristic \(p\) , Math. Ann. 272 (1985), 291-330. · Zbl 0553.14019
[12] N. M. Katz, “Serre-Tate local moduli” in Algebraic Surfaces (Orsay, 1976-1978) , Lecture Notes in Math. 868 , Springer, Berlin, 1981, 138-202. · Zbl 0477.14007
[13] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves , Ann. of Math. Stud. 108 , Princeton University Press, Princeton, 1985. · Zbl 0576.14026
[14] S. L. Kleiman, “The Picard scheme” in Fundamental Algebraic Geometry , Math. Surveys Monogr. 123 , Amer. Math. Soc., Providence, 2005, 235-321.
[15] W. E. Lang, Examples of liftings of surfaces and a problem in de Rham cohomology , Compos. Math. 97 (1995), 157-160. · Zbl 0877.14018
[16] Q. Liu, D. Lorenzini, and M. Raynaud, Néron models, Lie algebras, and reduction of curves of genus one , Invent. Math. 157 (2004), 455-518. · Zbl 1060.14037
[17] J. S. Milne, Étale Cohomology , Princeton Math. Ser. 33 , Princeton University Press, Princeton, 1980. · Zbl 0433.14012
[18] D. Mumford, Abelian Varieties , Tata Inst. Fund. Res. Stud. Math. 5 , Oxford University Press, London, 1970. · Zbl 0223.14022
[19] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory , 3rd ed., Ergeb. Math. Grenzgeb. (2) 34 , Springer, Berlin, 1994. · Zbl 0797.14004
[20] A. Pacheco and K. F. Stevenson, Finite quotients of the algebraic fundamental group of projective curves in positive characteristic , Pacific J. Math. 192 (2000), 143-158. · Zbl 0951.14016
[21] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes , Lecture Notes in Math. 119 , Springer, Berlin, 1970. · Zbl 0195.22701
[22] M. Schlessinger, Functors of Artin rings , Trans. Amer. Math. Soc. 130 (1968), 208-222. · Zbl 0167.49503
[23] J. H. Silverman, The Arithmetic of Elliptic Curves , 2nd ed., Grad. Texts in Math. 106 , Springer, Dordrecht, 2009. · Zbl 1194.11005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.