A Complex Ball Uniformization of the Moduli Space of Cubic Surfaces Via Periods of K3 Surfaces

I. Dolgachev, B. van Geemen, S. Kondo

Published 2003-10-21Version 1

In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K3 surfaces which are naturally associated to cubic surfaces. A similar uniformization is given for the covers of the moduli space corresponding to geometric markings of the Picard group or to the choice of a line on the surface. We also give a detailed description of the boundary components corresponding to singular surfaces.