Moduli of weighted stable marked cubic surfaces

Abstract

Let $Y(E_n)$ denote the moduli space of pairs $(S,B)$ where $S$ is a del Pezzo surface of degree $9-n$ and $B$ is the labeled (marked) sum of its finitely many lines. When $n=6$, $Y(E_6)$ is the classical moduli space of marked cubic surfaces dating back to the nineteenth century. We describe the compactifications of $Y(E_5)$ and $Y(E_6)$ by Kollár--Shepherd-Barron--Alexeev (KSBA) weighted stable pairs $(S,cB)$. There is a finite wall-and-chamber decomposition of the weight domain $\left(\frac{9-n}{N},1\right]$, and we explicitly identify this decomposition, as well as describe in detail the weighted stable pairs parameterized by the moduli spaces in each chamber. This generalizes the work of Hacking, Keel, and Tevelev constructing the moduli space and its universal family in the weight 1 case, and in particular yields a complete description of the fibers of this family.