## Journal of the Mathematical Society of Japan

### Kummer's quartics and numerically reflective involutions of Enriques surfaces

Shigeru MUKAI

#### Abstract

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

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

Dates
First available in Project Euclid: 26 January 2012

https://projecteuclid.org/euclid.jmsj/1327586981

Digital Object Identifier
doi:10.2969/jmsj/06410231

Mathematical Reviews number (MathSciNet)
MR2879743

Zentralblatt MATH identifier
1237.14046

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1327586981

#### References

• 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.