Kummer’s Quartics and Numerically Reflective Involutions of Enriques Surfaces

A (holomorphic) automorphism of an Enriques surface S is said to be numerically reflective if it acts on the cohomology group H(S,Q) by reflection. We shall show that there are two lattice-types of numerically reflective involutions, and describe one type geometrically in terms of curves of genus 2 and Göpel subgroups of their Jacobians. An automorphism of an Enriques surface S is numerically trivial if it acts on the cohomology group H(S,Q) ' Q trivially. By [MN] and [M06], there are three types of numerically trivial involutions. An involution of S is numerically reflective if it acts on H(S,Q) by reflection, that is, the eigenvalue −1 is of multiplicity one. In this article, we shall study numerically reflective involutions as the next case of the classification of involutions of Enriques surfaces. We first explain an example, with which we started our investigation. Let C be a (smooth projective) curve of gnus 2 and J = J(C) be its Jacobian variety. As is well known the quotient variety J(C)/{±1J} is realized as a quartic surface with 16 nodes in P, called Kummer’s quartic. The minimal resolution of J(C)/{±1J} is called the Jacobian Kummer surface of C and denoted by Km C. Let G ⊂ J(C)(2) be a Göpel subgroup such that the four associated nodes Ḡ ⊂ J(C)/{±1J} are linearly independent in P (Proposition 5.2). Then the equation of Kummer’s quartic J(C)/{±1J} ⊂ P referred to the four nodes has the form (1) a(xt + yz) + b(yt + zx) + c(zt + xy) + dxyzt +f(yt + zx)(zt + xy) + g(zt + xy)(xt + yz) + h(xt + yz)(yt + zx) = 0 for constants a, . . . , h ∈ C by Hutchinson[H01]. The standard Cremona transformation (x : y : z : t) 7→ (x−1 : y−1 : z−1 : t−1) Supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 17340006.