A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces
By Igor Dolgachev, Bert van Geemen, Shigeyuki Kondo. 2003
66 citations recorded by the source corpus.
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 K 3 surfaces which are naturally associated to cubic surfaces. A similar uniformization is given for different covers of the moduli space corresponding to geometric markings of the Picard group or a choice of a line on the surface. We also give a detailed description of the boundary components corresponding to singular surfaces.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7ea782cd75a5bd4a6796f26a937083ab9b1d88ea
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In Geometry and Topology
Citations in the corpus
- Special cycles on unitary Shimura varieties II: global theory (2009, background): “or example, in the case of cubic surfaces, the intersection of Im(ϕ) with the image of the KM-divisor Z(2) in M(k;4,1)C can be identified with the locus of cubic surfaces admitting Eckardt points, cf. [19], Thm. 8.10. Similarly, in the case of curves of genus 3, the intersection of Im(ϕ) with the ima”
- Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, background): “Some moduli of K3 surfaces with extra structure can be described as ball quotients: see for example [27].”
- Reflection groups in algebraic geometry (2006, result): “rfaces of degree 1. Independently such a construction was found in [57]. Finally, one can also reprove the result of Allcock-Carlson-Toledo by using K3 surfaces instead of intermediate jacobians (see [32]). Remark 10.3. All reflection groups arising in these complex ball uniformization constructions ”