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Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number

Part of: Families, fibrations Arithmetic problems. Diophantine geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 January 2009

Chad Schoen
Chad Schoen*
Affiliation:
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA (email: [email protected])
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Abstract

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Desingularized fiber products of semi-stable elliptic surfaces with Hetale3=0 are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of arithmetic degenerations of rigid Calabi–Yau threefolds.

Keywords

supersingular threefoldCalabi–Yau threefoldfiber product of elliptic surfaces

MSC classification

Secondary: 14D06: Fibrations, degenerations 14J32: Calabi-Yau manifolds 14G15: Finite ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009 , pp. 89 - 111
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Artin, M., Algebraization of formal moduli, I. Global analysis (Papers in honor of K. Kodaira) (University of Tokyo Press, Tokyo, 1969), 21–71.Google Scholar
[2]Artin, M., The implicit function theorem in algebraic geometry, in International Colloquim on Algebraic geometry (Tata Institute of Fundamental Research, Bombay, 1968) (Oxford University Press, London, 1969), 13–34.Google Scholar
[3]Artin, M., Algebraization of formal moduli, II. Existence of modifications, Ann. of Math. (2) 91 (1970), 88–135.CrossRefGoogle Scholar
[4]Artin, M., Théorèms de représentabilité pour les espaces algébriques (Presses de l’université de Montréal, Montréal, 1973).Google Scholar
[5]Beauville, A., Le nombre minimum de fibres singulières d’une courbe stable sur , in Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque, vol. 86, ed. L. Szpiro (Société Mathématique de France, Paris, 1981), 97–108.Google Scholar
[6]Beauville, A., Les familles stables de courbes elliptiques sur admettant quatre fibres singulières, C. R. Acad. Sci. Paris, Sér. I 294 (1982), 657–660.Google Scholar
[7]Deligne, P., La conjecture de Weil: II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.CrossRefGoogle Scholar
[8]Ekedahl, T., On non-liftable Calabi Yau threefolds, Preprint (2004), arXiv:math.AG/0306435 v2.Google Scholar
[9]Faltings, G., Schappacher, N., Wüstholz, G., et al. Rational points, third enlarged edition (Max-Planck-Institut für Mathematik, Bonn, 1992).CrossRefGoogle Scholar
[10]Grothendieck, A. and Dieudonné, J. A., Eléments de Géometrie Algébriques I, Grundlehren der Mathematischen Wissenschaften, Band 166 (Springer, Berlin, 1971).Google Scholar
[11]Grothendieck, A., Revêtements étales et groupe fondamental (SGA I), Lecture Notes in Mathematics, vol. 224 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[12]Hartshorne, R., Algebraic geometry (Springer, New York, 1977).CrossRefGoogle Scholar
[13]Hirokado, M., A non-liftable Calabi–Yau threefold in characteristic 3, Tohoku Math. J. 51 (1999), 479–487.CrossRefGoogle Scholar
[14]Hirokado, M., Calabi–Yau threefolds obtained as fiber products of elliptic and quasi-elliptic rational surfaces, J. Pure Appl. Algebra 162 (2001), 251–271.CrossRefGoogle Scholar
[15]Ito, H., On unirationality of extremal elliptic surfaces, Math. Ann. 310 (1998), 717–733.CrossRefGoogle Scholar
[16]Ito, H., On extremal elliptic surfaces in characteristic 2 and 3, Hiroshima Math. J. 32 (2002), 179–188.CrossRefGoogle Scholar
[17]Katz, N., Moments, monodromy, and perversity: a diophantine perspective (Princeton University Press, Princeton, NJ, 2005).Google Scholar
[18]Kleiman, S., Relative duality for quasi-coherent sheaves, Compositio Math. 41 (1980), 39–60.Google Scholar
[19]Knutson, D., Algebraic spaces, Lecture Notes in Mathematics, vol. 203 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[20]Lang, W., Extremal rational elliptic surfaces in characteristic p. I: Beauville surfaces, Math. Z. 207 (1991), 429–437.CrossRefGoogle Scholar
[21]Lewis, J., A survey of the Hodge conjecture (Les publications CRM, Montreal, 1991).Google Scholar
[22]Liu, Q., Algebraic geometry and arithmetic curves (Oxford Univeristy Press, Oxford, 2002).CrossRefGoogle Scholar
[23]Milne, J., Étale cohomology (Princeton University Press, Princeton, NJ, 1980).Google Scholar
[24]Miranda, R. and Persson, U., On extremal rational elliptic surfaces, Math. Z. 193 (1986), 537–558.CrossRefGoogle Scholar
[25]Moisezon, B., On n-dimensional compact varieties with n algebraically independent meromorphic functions. I, II, III, Amer. Math. Soc. Transl, Ser. 2 63 (1967), 51–177.Google Scholar
[26]Rudakov, A. N. and Safarevic, I. R., Supersingular K3 surfaces over fields of characteristic 2, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 848–869.Google Scholar
[27]Schoen, C., On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988), 177–199.CrossRefGoogle Scholar
[28]Schoen, C., Torsion in the cohomology of fiber products of elliptic surfaces, Preprint (2002).Google Scholar
[29]Schoen, C., Complex varieties for which the Chow group mod n is not finite, J. Algebraic Geom. 11 (2002), 41–100.CrossRefGoogle Scholar
[30]Schoen, C., Invariants of certain normal crossing surfaces, in preparation.Google Scholar
[31]Schröer, S., Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors, Composito Math. 140 (2004), 1579–1592.CrossRefGoogle Scholar
[32]Schütt, M., New examples of modular rigid Calabi–Yau threefolds, Collect. Math. 55 (2004), 219–228.Google Scholar
[33]Schweizer, A., Extremal elliptic surfaces in characteristic 2 and 3, Manuscripta Math. 102 (2000), 505–521.CrossRefGoogle Scholar
[34]Sernesi, E., Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, vol. 334 (Springer, New York, 2006).Google Scholar
[35]Shioda, T., On unirationality of supersingular surfaces, Math. Ann. 225 (1977), 155–159.CrossRefGoogle Scholar
[36]Silverman, J., The arithmetic of elliptic curves (Springer, New York, 1986).CrossRefGoogle Scholar
[37]Silverman, J., Advanced topics in the arithmetic of elliptic curves (Springer, New York, 1994).CrossRefGoogle Scholar