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

I. Dolgachev·B. van Geemen·S. Kondo
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Abstract

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.

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I. Dolgachev, B. van Geemen, S. Kondo. 2003-10-21. A Complex Ball Uniformization of the Moduli Space of Cubic Surfaces Via Periods of K3 Surfaces. https://arxiv.org/abs/math/0310342

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