Some surfaces with maximal Picard number

Beauville, Arnaud

doi:10.5802/jep.5

Some surfaces with maximal Picard number
[Quelques surfaces dont le nombre de Picard est maximal]

Beauville, Arnaud ¹

¹ Laboratoire J.-A. Dieudonné, UMR 7351 du CNRS, Université de Nice Parc Valrose, F-06108 Nice cedex 2, France

Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 101-116.

Résumé
Abstract

Le rang $ρ$ du groupe de Néron-Severi d’une variété projective lisse complexe est borné par le nombre de Hodge $h^{1, 1}$ . Les variétés satisfaisant à $ρ = h^{1, 1}$ ont des propriétés intéressantes, mais sont assez rares, particulièrement en dimension $2$ . Dans cette note nous analysons un certain nombre d’exemples, notamment ceux construits à partir de courbes à jacobienne spéciale.

For a smooth complex projective variety, the rank $ρ$ of the Néron-Severi group is bounded by the Hodge number $h^{1, 1}$ . Varieties with $ρ = h^{1, 1}$ have interesting properties, but are rather sparse, particularly in dimension $2$ . We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.

DOI : 10.5802/jep.5

Classification : 14J05, 14C22, 14C25
Keywords: Algebraic surfaces, Picard group, Picard number, curve correspondences, Jacobians
Mot clés : Surfaces algébriques, groupe de Picard, nombre de Picard, correspondances de courbes, jacobiennes

Affiliations des auteurs :

Beauville, Arnaud ¹

¹ Laboratoire J.-A. Dieudonné, UMR 7351 du CNRS, Université de Nice Parc Valrose, F-06108 Nice cedex 2, France

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Beauville, Arnaud. Some surfaces with maximal Picard number. Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 101-116. doi : 10.5802/jep.5. http://www.numdam.org/articles/10.5802/jep.5/

Bibliographie
Cité par

[Adl81] Adler, A. Some integral representations of ${PSL}_{2} (𝔽_{p})$ and their applications, J. Algebra, Volume 72 (1981) no. 1, pp. 115-145 | DOI | MR | Zbl

[Aok83] Aoki, N. On Some Arithmetic Problems Related to the Hodge Cycles on the Fermat Varieties, Math. Ann., Volume 266 (1983) no. 1, pp. 23-54 Erratum: ibid. 267 (1984) no. 4, p. 572 | DOI | MR | Zbl

[BD85] Beauville, A.; Donagi, R. La variété des droites d’une hypersurface cubique de dimension $4$ , C. R. Acad. Sci. Paris Sér. I Math., Volume 301 (1985) no. 14, pp. 703-706 | MR | Zbl

[BE87] Bertin, J.; Elencwajg, G. Configurations de coniques et surfaces avec un nombre de Picard maximum, Math. Z., Volume 194 (1987) no. 2, pp. 245-258 | DOI | MR | Zbl

[Bea13] Beauville, A. A tale of two surfaces (2013) (arXiv:1303.1910, to appear in the ASPM volume in honor of Y. Kawamata)

[Cat79] Catanese, F. Surfaces with $K^{2} = p_{g} = 1$ and their period mapping, Algebraic geometry (Copenhagen, 1978) (Lect. Notes in Math.), Volume 732, Springer, Berlin, 1979, pp. 1-29 | MR | Zbl

[CG72] Clemens, H. C.; Griffiths, P. A. The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), Volume 95 (1972), pp. 281-356 | MR | Zbl

[DK93] Dolgachev, I.; Kanev, V. Polar covariants of plane cubics and quartics, Advances in Math., Volume 98 (1993) no. 2, pp. 216-301 | DOI | MR | Zbl

[FSM13] Freitag, E.; Salvati Manni, R. Parametrization of the box variety by theta functions (2013) (arXiv:1303.6495) | MR

[Gri69] Griffiths, P. A. On the periods of certain rational integrals. I, II, Ann. of Math. (2), Volume 90 (1969), p. 460-495 & 496-541 | MR | Zbl

[HN65] Hayashida, T.; Nishi, M. Existence of curves of genus two on a product of two elliptic curves, J. Math. Soc. Japan, Volume 17 (1965), pp. 1-16 | MR | Zbl

[Hof91] Hoffmann, D. W. On positive definite Hermitian forms, Manuscripta Math., Volume 71 (1991) no. 4, pp. 399-429 | DOI | MR | Zbl

[Kat75] Katsura, T. On the structure of singular abelian varieties, Proc. Japan Acad., Volume 51 (1975) no. 4, pp. 224-228 | MR | Zbl

[Lan75] Lange, H. Produkte elliptischer Kurven, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1975) no. 8, pp. 95-108 | MR | Zbl

[LB92] Lange, H.; Birkenhake, C. Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, 302, Springer-Verlag, Berlin, 1992, pp. viii+435 | DOI | MR | Zbl

[Liv81] Livne, R. A. On certain covers of the universal elliptic curve, Harvard University (1981) (Ph.D. Thesis ProQuest LLC, Ann Arbor, MI) | MR

[Per82] Persson, U. Horikawa surfaces with maximal Picard numbers, Math. Ann., Volume 259 (1982) no. 3, pp. 287-312 | DOI | MR | Zbl

[Rou09] Roulleau, X. The Fano surface of the Klein cubic threefold, J. Math. Kyoto Univ., Volume 49 (2009) no. 1, pp. 113-129 | MR | Zbl

[Rou11] Roulleau, X. Fano surfaces with 12 or 30 elliptic curves, Michigan Math. J., Volume 60 (2011) no. 2, pp. 313-329 | DOI | MR | Zbl

[Sch11] Schütt, M. Quintic surfaces with maximum and other Picard numbers, J. Math. Soc. Japan, Volume 63 (2011) no. 4, pp. 1187-1201 http://projecteuclid.org/euclid.jmsj/1319721139 | MR | Zbl

[Shi69] Shioda, T. Elliptic modular surfaces. I, Proc. Japan Acad., Volume 45 (1969), pp. 786-790 | Zbl

[Shi79] Shioda, T. The Hodge conjecture for Fermat varieties, Math. Ann., Volume 245 (1979) no. 2, pp. 175-184 | DOI | MR | Zbl

[Shi81] Shioda, T. On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 28 (1981) no. 3, p. 725-734 (1982) | MR | Zbl

[Sil94] Silverman, J. H. Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Math., 151, Springer-Verlag, New York, 1994, pp. xiv+525 | DOI | MR | Zbl

[ST10] Stoll, M.; Testa, D. The surface parametrizing cuboids (2010) (arXiv:1009.0388)

[Tod80] Todorov, A. N. Surfaces of general type with $p_{g} = 1$ and $(K, K) = 1$ . I, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 1, pp. 1-21 | Numdam | MR | Zbl

[Tod81] Todorov, A. N. A construction of surfaces with $p_{g} = 1$ , $q = 0$ and $2 \leq (K^{2}) \leq 8$ . Counterexamples of the global Torelli theorem, Invent. Math., Volume 63 (1981) no. 2, pp. 287-304 | DOI | MR | Zbl

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