Nagoya Mathematical Journal

Kummer surfaces associated to (1,2)-polarized abelian surfaces

Afsaneh Mehran

Full-text: Open access

Abstract

The aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1,2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of type I2.

Article information

Source
Nagoya Math. J., Volume 202 (2011), 127-143.

Dates
First available in Project Euclid: 31 May 2011

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1306851593

Digital Object Identifier
doi:10.1215/00277630-1260477

Mathematical Reviews number (MathSciNet)
MR2804549

Zentralblatt MATH identifier
1223.14045

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces

Citation

Mehran, Afsaneh. Kummer surfaces associated to $(1,2)$ -polarized abelian surfaces. Nagoya Math. J. 202 (2011), 127--143. doi:10.1215/00277630-1260477. https://projecteuclid.org/euclid.nmj/1306851593


Export citation

References

  • [B] A. Beauville, Préliminaires sur les périodes des surfaces K3, Astérisque 126 (1985), 91–97.
  • [BL] C. Birkenhake and H. Lange, Complex Abelian Varieties, 2nd ed., Grundlehren Math. Wiss. 302, Springer, Berlin, 2004.
  • [H] R. W. H. T. Hudson, Kummer’s Quartic Surface, revised reprint of the 1905 original, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1990.
  • [K] J. H. Keum, Automorphisms of Jacobian Kummer surfaces, Compos. Math. 107 (1997), 269–288.
  • [M] A. Mehran, Double covers of Kummer surfaces, Manuscripta Math. 123 (2007), 205–235.
  • [Na] I. Naruki, On metamorphosis of Kummer surfaces, Hokkaido Math. J. 20 (1991), 407–415.
  • [Ni1] V. V. Nikulin, Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 278–293.
  • [Ni2] V. V. Nikulin, Finite groups of automorphisms of Kählerian K3 surfaces, Tr. Mosk. Mat. Obs. 38 (1979), 75–137.
  • [PŠŠ] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572.
  • [SI] T. Shioda and H. Inose, “On singular K3 surfaces” in Complex Analysis and Algebraic Geometry, Iwanami, Tokyo, 1977, 119–136.