no. 3
Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
[Calcul du rang de Mordell-Weil des variétés elliptiques de dimension trois et de la cohomologie des hypersurfaces singulières]
Hulek, Klaus1 ; Kloosterman, Remke2
1 Leibniz Universität Hannover Institut für Algebraische Geometrie Welfengarten 1 30167 Hannover (Germany)
2 Humboldt Universität zu Berlin Institut für Mathematik Unter den Linden 6 10099 Berlin (Germany)
Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1133-1179.

Dans cet article nous présentons une méthode pour calculer le rang d’une courbe elliptique générale sur le corps des fonctions rationnelles de deux variables. Nous réduisons ce problème au calcul de la cohomologie d’une hypersurface singulière dans un espace projectif pondéré de dimension quatre. Nous donnons alors une méthode de calcul de la cohomologie d’une certaine classe d’hypersurfaces singulières en étendant le travail de Dimca dans le cas des singularités isolées.

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

DOI : 10.5802/aif.2637
Classification : 14J30, 14J70, 32S20, 32S35, 32S50
Keywords: Mordel-Weil group of Elliptic threefolds, Cohomology of singular varieties, Mixed Hodge structures
Mot clés : Groupe de Mordell-Weil des variétés elliptiques de dimension trois, cohomologie des variétés singulières, structures de Hodge mixtes

Hulek, Klaus 1 ; Kloosterman, Remke 2

1 Leibniz Universität Hannover Institut für Algebraische Geometrie Welfengarten 1 30167 Hannover (Germany)
2 Humboldt Universität zu Berlin Institut für Mathematik Unter den Linden 6 10099 Berlin (Germany) 
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Hulek, Klaus; Kloosterman, Remke. Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces. Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1133-1179. doi : 10.5802/aif.2637. https://aif.centre-mersenne.org/articles/10.5802/aif.2637/

[1] Batyrev, V. V. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., Volume 3 (1994), pp. 493-535 | MR | Zbl

[2] Behrens, N. Calabi-Yau 3-Varietäten mit elliptischen Faserungen über Del Pezzo-Flächen, Diplomarbeit, Leibniz Universität Hannover, Hannover, 2006

[3] Clemens, C. H. Double solids, Adv. in Math., Volume 47 (1983), pp. 107-230 | DOI | MR | Zbl

[4] Cox, D. A. The Noether-Lefschetz locus of regular elliptic surfaces with section and p g ≥2, Amer. J. Math., Volume 112 (1990), pp. 289-329 | DOI | MR | Zbl

[5] Cynk, S. Defect of a nodal hypersurface, Manuscripta Math., Volume 104 (2001), pp. 325-331 | DOI | MR | Zbl

[6] Deligne, P.; Dimca, A. Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. École Norm. Sup. (4), Volume 23 (1990), pp. 645-656 | Numdam | MR | Zbl

[7] Dimca, A. Topics on real and complex singularities, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1987 | MR | Zbl

[8] Dimca, A. Betti numbers of hypersurfaces and defects of linear systems, Duke Math. J., Volume 60 (1990), pp. 285-298 | DOI | MR | Zbl

[9] Dimca, A. Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992 | MR | Zbl

[10] Dimca, A.; Saito, M.; Wotzlaw, L. A generalization of Griffiths’ theorem on rational integrals, II, 2007 (preprint available at arXiv:math/0702105v6) | Zbl

[11] van Geemen, B.; Werner, J. Nodal quintics in P 4 , Arithmetic of complex manifolds (Erlangen, 1988) (Lecture Notes in Math.), Volume 1399, Springer, Berlin, 1989, pp. 48-59 | MR | Zbl

[12] Griffiths, P. A. On the periods of certain rational integrals. II, Ann. of Math. (2), Volume 90 (1969), pp. 496-541 | DOI | MR | Zbl

[13] Grooten, M.; Steenbrink, J. H. M. Defect and Hodge numbers of hypersurfaces, 2007 (in preperation)

[14] de Jong, T.; Pfister, G. Local analytic geometry, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 2000 | MR | Zbl

[15] Kloosterman, R. Elliptic K3 surfaces with geometric Mordell-Weil rank 15, Canad. Math. Bull., Volume 50 (2007), pp. 215-226 | DOI | MR | Zbl

[16] Kloosterman, R. Higher Noether-Lefschetz loci of elliptic surfaces, J. Differential Geom., Volume 76 (2007), pp. 293-316 | MR | Zbl

[17] Kloosterman, R. On the classification of degree 1 elliptic threefolds with constant j-invariant, 2008 (preprint available at arxiv:0812.3014) | Zbl

[18] Kloosterman, R. A different method to calculate the rank of an elliptic threefold, to appear in Rocky Mountain J. Math., available at arxiv:0812.3222 | Zbl

[19] Miranda, R. Smooth models for elliptic threefolds, The birational geometry of degenerations (Cambridge, Mass., 1981) (Progr. Math.), Volume 29, Birkhäuser Boston, Mass., 1983, pp. 85-133 | MR | Zbl

[20] Miranda, R. The basic theory of elliptic surfaces, ETS Editrice, Pisa, 1989 | MR | Zbl

[21] Oguiso, K.; Shioda, T. The Mordell-Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul, Volume 40 (1991), pp. 83-99 | MR | Zbl

[22] Peters, C. A. M.; Steenbrink, J. H. M. Mixed Hodge structures, Ergebnisse der Mathematik, 52, Springer, 2008 | MR | Zbl

[23] Rams, S. Defect and Hodge numbers of hypersurfaces, 2007 (preprint available at arXiv: math/0702114v1) | MR | Zbl

[24] Schoen, C. Algebraic cycles on certain desingularized nodal hypersurfaces, Math. Ann., Volume 270 (1985), pp. 17-27 | DOI | EuDML | MR | Zbl

[25] Steenbrink, J. H. M. Intersection form for quasi-homogeneous singularities, Compositio Math., Volume 34 (1977), pp. 211-223 | EuDML | Numdam | MR | Zbl

[26] Steenbrink, J. H. M. Adjunction conditions for one-forms on surfaces in projective three-space, Singularities and computer algebra (London Math. Soc. Lecture Note Ser.), Volume 324, Cambridge Univ. Press, Cambridge, 2006, pp. 301-314 | MR | Zbl

[27] Vosion, C. Hodge theory and complex algebraic geometry. II, Cambridge Studies in Advanced Mathematics, 77, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[28] Wazir, R. Arithmetic on elliptic threefolds, Compos. Math., Volume 140 (2004), pp. 567-580 | DOI | MR | Zbl

[29] Werner, J. Kleine Auflösungen spezieller dreidimensionaler Varietäten, Bonner Mathematische Schriften, 186, Universität Bonn Mathematisches Institut, Bonn, 1987 (Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1987) | MR | Zbl

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