Some surfaces with maximal Picard number

Arnaud Beauville[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 (2014)

  • Volume: 1, page 101-116
  • ISSN: 2270-518X

Abstract

top
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.

How to cite

top

Beauville, Arnaud. "Some surfaces with maximal Picard number." Journal de l’École polytechnique — Mathématiques 1 (2014): 101-116. <http://eudml.org/doc/275663>.

@article{Beauville2014,
abstract = {For a smooth complex projective variety, the rank $\rho $ of the Néron-Severi group is bounded by the Hodge number $h^\{1,1\}$. Varieties with $\rho =h^\{1,1\}$ 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.},
affiliation = {Laboratoire J.-A. Dieudonné, UMR 7351 du CNRS, Université de Nice Parc Valrose, F-06108 Nice cedex 2, France},
author = {Beauville, Arnaud},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Algebraic surfaces; Picard group; Picard number; curve correspondences; Jacobians; algebraic surfaces},
language = {eng},
pages = {101-116},
publisher = {École polytechnique},
title = {Some surfaces with maximal Picard number},
url = {http://eudml.org/doc/275663},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Beauville, Arnaud
TI - Some surfaces with maximal Picard number
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 101
EP - 116
AB - For a smooth complex projective variety, the rank $\rho $ of the Néron-Severi group is bounded by the Hodge number $h^{1,1}$. Varieties with $\rho =h^{1,1}$ 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.
LA - eng
KW - Algebraic surfaces; Picard group; Picard number; curve correspondences; Jacobians; algebraic surfaces
UR - http://eudml.org/doc/275663
ER -

References

top
  1. A. Adler, Some integral representations of PSL 2 ( 𝔽 p ) and their applications, J. Algebra 72 (1981), 115-145 Zbl0479.20017MR634619
  2. N. Aoki, On Some Arithmetic Problems Related to the Hodge Cycles on the Fermat Varieties, Math. Ann. 266 (1983), 23-54 Zbl0506.14030MR722926
  3. A. Beauville, R. Donagi, La variété des droites d’une hypersurface cubique de dimension  4 , C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 703-706 Zbl0602.14041MR818549
  4. J. Bertin, G. Elencwajg, Configurations de coniques et surfaces avec un nombre de Picard maximum, Math. Z. 194 (1987), 245-258 Zbl0612.14031MR876234
  5. A. Beauville, A tale of two surfaces, (2013) 
  6. F. Catanese, Surfaces with K 2 = p g = 1 and their period mapping, Algebraic geometry (Copenhagen, 1978) 732 (1979), 1-29, Springer, Berlin Zbl0423.14019MR555688
  7. H. C. Clemens, P. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281-356 Zbl0214.48302MR302652
  8. I. Dolgachev, V. Kanev, Polar covariants of plane cubics and quartics, Advances in Math. 98 (1993), 216-301 Zbl0791.14013MR1213725
  9. E. Freitag, R. Salvati Manni, Parametrization of the box variety by theta functions, (2013) Zbl1285.11076MR3078079
  10. P. A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495 & 496-541 Zbl0215.08103MR260733
  11. T. Hayashida, M. Nishi, Existence of curves of genus two on a product of two elliptic curves, J. Math. Soc. Japan 17 (1965), 1-16 Zbl0132.41701MR201434
  12. D. W. Hoffmann, On positive definite Hermitian forms, Manuscripta Math. 71 (1991), 399-429 Zbl0729.11020MR1104993
  13. T. Katsura, On the structure of singular abelian varieties, Proc. Japan Acad. 51 (1975), 224-228 Zbl0321.14021MR376695
  14. H. Lange, Produkte elliptischer Kurven, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1975), 95-108 Zbl0317.14022MR506297
  15. H. Lange, C. Birkenhake, Complex abelian varieties, 302 (1992), Springer-Verlag, Berlin Zbl1056.14063MR1217487
  16. R. A. Livne, On certain covers of the universal elliptic curve, (1981) MR2936887
  17. U. Persson, Horikawa surfaces with maximal Picard numbers, Math. Ann. 259 (1982), 287-312 Zbl0466.14010MR661198
  18. X. Roulleau, The Fano surface of the Klein cubic threefold, J. Math. Kyoto Univ. 49 (2009), 113-129 Zbl1207.14045MR2531132
  19. X. Roulleau, Fano surfaces with 12 or 30 elliptic curves, Michigan Math. J. 60 (2011), 313-329 Zbl1225.14029MR2825265
  20. M. Schütt, Quintic surfaces with maximum and other Picard numbers, J. Math. Soc. Japan 63 (2011), 1187-1201 Zbl1232.14022MR2855811
  21. T. Shioda, Elliptic modular surfaces. I, Proc. Japan Acad. 45 (1969), 786-790 Zbl0198.26301
  22. T. Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), 175-184 Zbl0403.14007MR552586
  23. T. Shioda, On the Picard number of a Fermat surface, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 725-734 (1982) Zbl0567.14021MR656049
  24. J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, 151 (1994), Springer-Verlag, New York Zbl0911.14015MR1312368
  25. M. Stoll, D. Testa, The surface parametrizing cuboids, (2010) 
  26. A. N. Todorov, Surfaces of general type with p g = 1 and ( K , K ) = 1 . I, Ann. Sci. École Norm. Sup. (4) 13 (1980), 1-21 Zbl0478.14030MR584080
  27. A. N. Todorov, A construction of surfaces with p g = 1 , q = 0 and 2 ≤ ( K 2 ) ≤ 8 . Counterexamples of the global Torelli theorem, Invent. Math. 63 (1981), 287-304 Zbl0457.14016MR610540

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.