On the Number of Enriques Quotients of a K3 Surface

Abstract

§0. Introduction A K3 surface X is a compact complex surface with KX ∼ 0 and H(X, OX) = 0. An Enriques surface is a compact complex surface with H(Y,OY ) = H(Y,OY ) = 0 and 2KY ∼ 0. The universal covering of an Enriques surface is a K3 surface. Conversely every quotient of a K3 surface by a free involution is an Enriques surface. Here a free involution is an automorphism of order 2 without any fixed points. The moduli space of Enriques surfaces is constructed using the periods of their covering K3 surfaces. Precisely speaking, an Enriques surface determines a lattice-polarized K3 surface and vice versa, so that the moduli space of Enriques surfaces can be described by the moduli space of lattice-polarized K3 surfaces. We note that even if we do not fix any polarization on Enriques surfaces, their covering K3 surfaces automatically have a lattice-polarization. Then, what happens if we drop the lattice-polarization of the covering K3 surface? We will call two Enriques quotients of a K3 surface distinct if they are not isomorphic to each other as varieties. In his paper [3], Kondo discovered a K3 surface with two distinct Enriques quotients. He computed the automorphism groups of the two quotients. Since then, as far as the author knows, no other examples have been found. In this paper we investigate this phenomenon. We show that K3 surfaces with more than one distinct Enriques quotients have 9-dimensional components (neither irreducible nor closed) in the period domain. Moreover we compute the exact number of distinct Enriques quotients at a very general point of