Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T23:11:39.086Z Has data issue: false hasContentIssue false
  • English
  • Français

Enriques Surfaces Covered by Jacobian Kummer Surfaces

Published online by Cambridge University Press:  11 January 2016

Hisanori Ohashi
Hisanori Ohashi*
Affiliation:
Research Institute for Mathematical SciencesKyoto University, Sakyo-ku, Kyoto 606-8502, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that Aut(X) is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.

Keywords

14J28
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 195 , 2009 , pp. 165 - 186
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Barth, W. and Peters, C., Automorphisms of Enriques surfaces, Invent. math., 73 (1983), 383–411.CrossRefGoogle Scholar
[2] Bourbaki, N., Eléments de Mathématique. Groupes et Algèbres de Lie, Chap. IV, V et VI.Google Scholar
[3] Conway, J. H. and Sloane, N. J. A., Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften 290, Springer-Verlag, 1999.CrossRefGoogle Scholar
[4] Dolgachev, I. V. and Keum, J. H., Birational automorphisms of quartic hessian surfaces, Trans. Amer. Math. Soc., 354 (2002), 3031–3057.CrossRefGoogle Scholar
[5] Hutchinson, J. I., The Hessian of the cubic surface, Bull. Amer. Math. Soc., 5 (1899), 282–292.Google Scholar
[6] Hutchinson, J. I., The Hessian of the cubic surface, II, Bull. Amer. Math. Soc., 6 (1899), 328–337.CrossRefGoogle Scholar
[7] Hutchinson, J. I., On some birational transformations of the Kummer surfaces into itself, Bull. Amer. Math. Soc. (2), 7 (1901), 211–217.CrossRefGoogle Scholar
[8] Klein, F., Ueber Configurationen, Welche der Kummer’schen Fl¨ache Zugleich Eingeschrieben und Umgeschrieben Sind, Math. Ann., 27 (1886), 106–142.Google Scholar
[9] Gonzalez-Dorrego, M. R., (16,6) configurations and geometry of Kummer surfaces in P3 , Mem. Amer. Math. Soc. 107, 1994.Google Scholar
[10] Keum, J. H., Every algebraic Kummer surface is the K3-cover of an Enriques surface, Nagoya Math. J., 118 (1990), 99–110.CrossRefGoogle Scholar
[11] Keum, J. H., Automorphisms of Jacobian Kummer surfaces, Compositio Math., 107 (1997), 269–288.CrossRefGoogle Scholar
[12] Kondo, S., The automorphism group of a generic Jacobian Kummer surface, J. Algebraic Geometry, 7 (1998), 589–609.Google Scholar
[13] Mukai, S., Kummer’s quartics and numerically reflective involutions of Enriques surfaces, RIMS Preprint, 1633.Google Scholar
[14] Nikulin, V. V., An analogue of the Torelli theorem for Kummer surfaces of Jacobians (English translation), Math. USSR Izv., 8 (1974), 21–41.CrossRefGoogle Scholar
[15] Nikulin, V. V., Integral symmetric bilinear forms and some of their applications (English translation), Math. USSR Izv., 14 (1980), 103–167.CrossRefGoogle Scholar
[16] Ohashi, H., On the number of Enriques quotients of a K3 surface, Publ. Res. Inst. Math. Sci., 43 (2007), 181–200.Google Scholar
[17] Ohashi, H., Counting Enriques quotients of a K3 surface, RIMS Preprint, 1609.Google Scholar