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.
A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces / I. Dolgachev, B. van Geemen, S. Kondo. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - 2005:588(2005), pp. 99-148. [10.1515/crll.2005.2005.588.99]
A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces
B. van GeemenSecondo
;
2005
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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