Journal of the Mathematical Society of Japan

Kummer's quartics and numerically reflective involutions of Enriques surfaces

Shigeru MUKAI

Full-text: Open access


A (holomorphic) involution σ of an Enriques surface S is said to be numerically reflective if it acts on the cohomology group H2(S, Q) as a reflection. We show that the invariant sublattice H(S, σ; Z) of the anti-Enriques lattice H-(S, Z) under the action of σ is isomorphic to either 〈-4〉 ⊥ U(2) ⊥ U(2) or 〈-4〉 ⊥ U(2) ⊥ U. Moreover, when H(S, σ; Z) is isomorphic to 〈-4〉 ⊥ U(2) ⊥ U(2), we describe (S, σ) geometrically in terms of a curve of genus two and a Göpel subgroup of its Jacobian.

Article information

J. Math. Soc. Japan, Volume 64, Number 1 (2012), 231-246.

First available in Project Euclid: 26 January 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14K10: Algebraic moduli, classification [See also 11G15] 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]

Enriques surface Kummer surface period


MUKAI, Shigeru. Kummer's quartics and numerically reflective involutions of Enriques surfaces. J. Math. Soc. Japan 64 (2012), no. 1, 231--246. doi:10.2969/jmsj/06410231.

Export citation


  • W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, 1984.
  • C. Birkenhake and H. Lange, Complex Abelian Varieties, Springer-Verlag, 2004.
  • I. V. Dolgachev and J. H. Keum, Birational automorphisms of quartic Hessian surfaces, Trans. Amer. Math. Soc., 354 (2002), 3031–3057.
  • G. van der Geer, Hilbert Modular Surfaces, 2nd ed., Springer-Verlag, 1988.
  • F. Hirzebruch, The ring of Hilbert modular forms for real quadratic fields of small discriminant, “Modular functions of one variable VI”, Lecture Notes in Math., 627 (1977), 287–323.
  • J. I. Hutchinson, The Hessian of the cubic surface, Bull. Amer. Math. Soc., 5 (1899), 282–292: II, ibid, 6 (1900), 328–337.
  • J. I. Hutchinson, On some birational transformations of the Kummer surface into itself, Bull. Amer. Math. Soc., 7 (1901), 211–217.
  • J. H. Keum, Every algebraic Kummer surface is the $K3$-cover of an Enriques surface, Nagoya Math. J., 118 (1990), 99–110.
  • D. R. Morrison, On $K3$ surfaces with large Picard number, Invent. Math., 75 (1984), 105–121.
  • S. Mukai, Numerically trivial involutions of Kummer type of an Enriques surface, Kyoto J. Math., 50 (2010), 889–902.
  • S. Mukai and Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on the cohomology groups, Invent. Math., 77 (1984), 383–397.
  • V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Izv. Akad.Nauk SSSR Ser. Mat., 43 (1979), 111–177 (Russian); Math. USSR-Izv., 14 (1980), 103–167.
  • H. Ohashi, Enriques surfaces covered by Jacobian Kummer surfaces, Nagoya Math. J., 195 (2009), 165–186.
  • T. Shioda, The period map of abelian surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 25 (1978), 47–59.