Journal of the Mathematical Society of Japan

Kummer's quartics and numerically reflective involutions of Enriques surfaces

Shigeru MUKAI

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1327586981

Digital Object Identifier
doi:10.2969/jmsj/06410231

Mathematical Reviews number (MathSciNet)
MR2879743

Zentralblatt MATH identifier
1237.14046

Subjects
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]

Keywords
Enriques surface Kummer surface period

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


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