Moduli of cubic surfaces and their anticanonical divisors

TL;DR

This paper studies the moduli space of cubic surfaces with anticanonical divisors, describing its compactifications via GIT, analyzing stability conditions, and characterizing singularities of stable pairs.

Contribution

It provides a comprehensive description of all GIT-based compactifications of the moduli space, including stability conditions and singularity characterizations.

Findings

Identified all walls in the stability space and related them to singularities.

Characterized stable and polystable pairs based on their singularities.

Described how stability varies with the weight parameter.

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.