Kummer's quartics and numerically reflective involutions of Enriques surfaces

摘要

A (holoraorphic) involution σ of an Enriques surface S is said to be numerically reflective if it acts on the cohomology group H~2(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) X 1/(2) X U{2) or (-4) X U(2) X U. Moreover, when H(S, σ; Z) is isomorphic to (-4) ⊥ U(2) X U(2), we describe (S,σ) geometrically in terms of a curve of genus two and a Gopel subgroup of its Jacobian.