On the extendability of elliptic surfaces of rank two and higher

Lopez, Angelo Felice; Muñoz, Roberto; Sierra, José Carlos

doi:10.5802/aif.2432

On the extendability of elliptic surfaces of rank two and higher
[Sur les extensions des surfaces elliptiques de rang supérieur ou égal à deux]

Lopez, Angelo Felice ¹ ; Muñoz, Roberto ² ; Sierra, José Carlos ³

¹ Universitá di Roma Tre Dipartimento di Matematica Largo San Leonardo Murialdo 1 00146 Roma (Italy)
² Universidad Rey Juan Carlos Departamento de Matemática Aplicada 28933 Móstoles Madrid (Spain)
³ Universidad Complutense de Madrid Facultad de Ciencias Matemáticas Departamento de Álgebra 28040 Madrid (Spain)

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 311-346.

Résumé
Abstract

On étudie les variétés de dimension trois $X \subset ℙ^{r}$ qui ont comme section hyperplane une surface lisse avec une fibration elliptique. On prouve d’abord un théorème général sur les plongements possibles de ces surfaces de nombre de Picard égal à deux. Dans un deuxième temps, on prouve des résultats plus précis pour les fibrations de Weierstrass de rang supérieur ou égal à deux. En particulier, on prouve qu’une fibration de Weierstrass de rang deux qui n’est pas une surface K3 n’est pas une section hyperplane d’une variété de dimension trois localement intersection complète. On donne, de plus, des conditions sous lesquelles, pour beaucoup des plongements de fibrations de Weierstrass de rang quelconque, toute variété de dimension trois comme ci-dessus est un cône.

We study threefolds $X \subset ℙ^{r}$ having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.

MR Zbl

DOI : 10.5802/aif.2432

Classification : 14J30, 14J27, 11G05
Keywords: Elliptic surfaces, hyperplane sections, Mori fiber spaces
Mots-clés : surfaces elliptiques, sections hyperplanes, fibration de Mori

Affiliations des auteurs :

Lopez, Angelo Felice ¹ ; Muñoz, Roberto ² ; Sierra, José Carlos ³

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     journal = {Annales de l'Institut Fourier},
     pages = {311--346},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Lopez, Angelo Felice; Muñoz, Roberto; Sierra, José Carlos. On the extendability of elliptic surfaces of rank two and higher. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 311-346. doi : 10.5802/aif.2432. https://aif.centre-mersenne.org/articles/10.5802/aif.2432/

Bibliographie
Cité par

[1] Altman, A.; Kleiman, S. Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Volume 146, Springer-Verlag, Berlin-New York, 1970 | MR | Zbl

[2] Arbarello, E.; Sernesi, E. Petri’s approach to the study of the ideal associated to a special divisor, Invent. Math., Volume 49 (1978), pp. 99-119 | DOI | MR | Zbl

[3] Badescu, L.; Springer Infinitesimal deformations of negative weights and hyperplane sections, Algebraic geometry, Volume 1417 (1990), pp. 1-22 | MR | Zbl

[4] Barth, W.; Peters, C.; van de Ven, A. Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 4, Springer-Verlag, Berlin-New York, 1984 | MR | Zbl

[5] Beltrametti, M. C.; Sommese, A. J. The adjunction theory of complex projective varieties, de Gruyter Expositions in Mathematics, Volume 16, Walter de Gruyter & Co., Berlin, 1995 | MR | Zbl

[6] Bertram, A.; Ein, L.; Lazarsfeld, R.; Springer Surjectivity of Gaussian maps for line bundles of large degree on curves, Algebraic geometry (Lecture Notes in Math.), Volume 1479 (1991), pp. 15-25 | MR | Zbl

[7] Brown, G.; Corti, A.; Zucconi, F. Birational geometry of 3-fold Mori fiber spaces (2004), pp. 235-275 | MR | Zbl

[8] Catanese, F.; Franciosi, M. Divisors of small genus on algebraic surfaces and projective embeddings, Israel Math. Conf. Proc., Volume 9 (1996), pp. 109-140 | MR | Zbl

[9] Ciliberto, C.; Lopez, A. F.; Miranda, R. Projective degenerations of $K 3$ surfaces, Gaussian maps, and Fano threefolds, Invent. Math., Volume 114 (1993), pp. 641-667 | DOI | MR | Zbl

[10] Ciliberto, C.; Lopez, A. F.; Miranda, R. Classification of varieties with canonical curve section via Gaussian maps on canonical curves, Amer. J. Math., Volume 120 (1998), pp. 1-21 | DOI | MR | Zbl

[11] Cox, D. A. The Noether-Lefschetz locus of regular elliptic surfaces with section and $p_{g} \geq 2$ , Amer. J. Math., Volume 112 (1990), pp. 289-329 | DOI | MR | Zbl

[12] D’Souza, H. Threefolds whose hyperplane sections are elliptic surfaces, Pacific J. Math., Volume 134 (1988), pp. 57-78 | MR | Zbl

[13] Elkik, R. Rationalité des singularités canoniques, Invent. Math., Volume 64 (1981), pp. 1-6 | DOI | MR | Zbl

[14] Fania, M. L.; D’Souza, H. Varieties whose surface sections are elliptic, Tohoku Math. J., Volume 42 (1990), pp. 457-474 | DOI | MR | Zbl

[15] Fujita, T. On singular del Pezzo varieties, Algebraic geometry (Lecture Notes in Math.), Volume 1417 (1990), pp. 117-128 | MR | Zbl

[16] Gherardelli, F. Un teorema di Lefschetz sulle intersezioni complete, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur., Volume 28 (1960), pp. 610-614 | MR | Zbl

[17] Giraldo, L.; Lopez, A. F.; Muñoz, R. On the existence of Enriques-Fano threefolds of index greater than one, J. Algebraic Geom., Volume 13 (2004), pp. 143-166 | DOI | MR | Zbl

[18] Green, M. Koszul cohomology and the geometry of projective varieties, J. Differ. Geom., Volume 19 (1984), pp. 125-171 | MR | Zbl

[19] Harris, J. Algebraic geometry. A first course, Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1992 | MR | Zbl

[20] Hartshorne, R. Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977 | MR | Zbl

[21] Ionescu, P. Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc., Volume 99 (1986), pp. 457-472 | DOI | MR | Zbl

[22] Iskovskih, V. A. Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat., Volume 41 (1977), p. 516-562, 717 English translation in Math. USSR-Izvestija, 11 (1977), 485–527 | MR | Zbl

[23] Iskovskih, V. A. Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat., Volume 42 (1978), pp. 506-549 English translation in Math. USSR-Izvestija, 12 (1978), 469–506 | MR | Zbl

[24] Kawamata, Y.; Matsuda, K.; Matsuki, K. Introduction to the minimal model problem, Algebraic geometry (Adv. Stud. Pure Math.), Volume 10 (1987), pp. 283-360 | MR | Zbl

[25] Kleiman, S. Les théorèmes de finitude pour le foncteur de Picard, Lecture Notes in Mathematics, Volume 225 (1971), pp. 616-666 (Exposé XIII) | Zbl

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

[27] Knutsen, A. L. Smooth curves on projective $K 3$ surfaces, Math. Scand., Volume 90 (2002), pp. 215-231 | MR | Zbl

[28] Knutsen, A. L.; Lopez, A. F. Surjectivity of Gaussian maps for curves on Enriques surfaces, Adv. Geom., Volume 7 (2007), pp. 215-247 | DOI | MR | Zbl

[29] Knutsen, A. L.; Lopez, A. F.; Muñoz, R. On the extendability of projective surfaces and a genus bound for Enriques-Fano threefolds (2006) (Preprint)

[30] Kollár, J.; Mori, S. Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[31] Kovács, S. J. The cone of curves of a $K 3$ surface, Math. Ann., Volume 300 (1994), pp. 681-691 | DOI | MR | Zbl

[32] Lanteri, A.; Maeda, H. Elliptic surfaces and ample vector bundles, Pacific J. Math., Volume 200 (2001), pp. 147-157 | DOI | MR | Zbl

[33] Lazarsfeld, R. A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces, World Sci. Publishing, Teaneck, NJ, 1989, pp. 500-559 (Trieste, 1987) | MR | Zbl

[34] Lazarsfeld, R. Positivity in Algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, 48, Springer-Verlag, Berlin, 2004 | MR | Zbl

[35] Lopez, A. F. Noether-Lefschetz theory and the Picard group of projective surfaces, Mem. Amer. Math. Soc., Volume 89 (1991) no. 438, pp. 100p. | MR | Zbl

[36] L’vovsky, S. Extensions of projective varieties and deformations. I, Michigan Math. J., Volume 39 (1992), pp. 41-51 | DOI | MR | Zbl

[37] Matsuki, K. Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002 | MR | Zbl

[38] Miranda, R. The moduli of Weierstrass fibrations over $ℙ^{1}$ , Math. Ann., Volume 255 (1981), pp. 379-394 | DOI | MR | Zbl

[39] Miranda, R. The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, Pisa, 1989 | MR | Zbl

[40] Moishezon, B. G. Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat., Volume 31 (1967), pp. 225-268 English translation in Math. USSR-Izvestija, 1 (1967), 209–251 | MR | Zbl

[41] Mori, S. Threefolds whose canonical bundles are not numerically effective, Ann. of Math., Volume 116 (1982), pp. 133-176 | DOI | MR | Zbl

[42] Mori, S.; Mukai, S. Classification of Fano $3$ -folds with $B_{2} \geq 2$ , Manuscripta Math., Volume 36 (1981/82), pp. 147-162 Erratum: “Classification of Fano 3-folds with $B_{2} \geq 2$ ". Manuscripta Math., 110 (2003), 407 | DOI | MR | Zbl

[43] Nagata, M. On rational surfaces. I. Irreducible curves of arithmetic genus $0$ or $1$ , Mem. Coll. Sci. Univ. Kyoto Ser. A Math., Volume 32 (1960), pp. 351-370 | MR | Zbl

[44] Namikawa, Y. Smoothing Fano 3-folds, J. Algebraic Geom., Volume 6 (1997), pp. 307-324 | MR | Zbl

[45] Paoletti, R. Free pencils on divisors, Math. Ann., Volume 303 (1995), pp. 109-123 | DOI | MR | Zbl

[46] Prokhorov, Yu. G. The degree of Fano threefolds with canonical Gorenstein singularities, Mat. Sb., Volume 196 (2005), pp. 81-122 English translation: Sb. Math., 196 (2005), 77–114 | MR | Zbl

[47] Shokurov, V. V. The existence of a line on Fano varieties, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979), p. 922-964, 968 English translation: Math. USSR-Izv., 15 (1979), 173–209 | MR | Zbl

[48] Shokurov, V. V. Smoothness of a general anticanonical divisor on a Fano variety, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979), pp. 430-441 English translation: Math. USSR-Izv., 14 (1980), 395–405 | MR | Zbl

[49] Wahl, J. Introduction to Gaussian maps on an algebraic curve, Complex Projective Geometry. Trieste-Bergen 1989, Lond. Math. Soc. Lect. Note Ser. 179, Cambridge Univ. Press, Cambridge 1992, 304-323

[50] Zak, F. L. Some properties of dual varieties and their applications in projective geometry, Algebraic geometry (Lecture Notes in Math.), Volume 1479 (1991), pp. 273-280 | MR | Zbl

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