KSBA moduli spaces of cubic surfaces with a marked line

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.