Moduli of cubic surfaces and their anticanonical divisors

Abstract

We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber structure. We describe all the walls and relate their value to the worst singularities appearing in the compactification locus. Furthermore, we give a complete characterization of stable and polystable pairs in terms of their singularities for each of the compactifications considered.

23 pages, 2 figures, 4 tables. v3: Paper reorganised and shortened. Introduction improved. Same results. Final version. To appear in Rev. Mat. Complut

References in corpus (1)

• Variations of geometric invariant quotients for pairs, a computational approach

Cited by in corpus (3)

• Applications of the moduli continuity method to log K-stable pairs
• Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones
• The moduli space of cubic surface pairs via the intermediate Jacobians of Eckardt cubic threefolds