Kummer's quartics and numerically reflective involutions of Enriques surfaces

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.