KSBA moduli spaces of cubic surfaces with a marked line

TL;DR

This paper extends the KSBA moduli space framework for cubic surfaces by incorporating a marked line with a distinct weight, leading to new compactifications and explicit chamber decompositions.

Contribution

It introduces nonuniform weights for the marked line in KSBA stable pairs, expanding the understanding of cubic surface moduli spaces.

Findings

Explicit finite wall-and-chamber decomposition of the weight domain.

New KSBA coarse moduli spaces for cubic surfaces with a marked line.

Descriptions of stable pairs in each chamber with nonuniform weights.

Abstract

The moduli space of cubic surfaces and its compactifications are classical and date back to the mid-nineteenth century. Recently, Schock described compactifications of moduli spaces of fully marked cubic surfaces with their 27 lines via Koll\'ar--Shepherd-Barron--Alexeev (KSBA) weighted stable pairs where the 27 lines are uniformly weighted. Furthermore, he provided an explicit finite wall-and-chamber decomposition of the weight domain, together with explicit descriptions of the weighted stable pairs parameterized by the moduli spaces in each chamber. We extend this work by describing compactifications of moduli spaces of cubic surfaces with a marked line via KSBA stable pairs with nonuniform weights, in which the marked line is weighted differently from the other 26 lines. In particular, we provide an explicit finite wall-and-chamber decomposition of the weight domain, yielding new KSBA coarse moduli spaces that have not previously been studied. We also give explicit descriptions of these nonuniform weighted stable pairs parameterized by the moduli spaces in each chamber.