Fineness and smoothness of a KSBA moduli of marked cubic surfaces

Abstract

By work of Gallardo-Kerr-Schaffler, it is known that Naruki's compactification of the moduli space of marked cubic surfaces is isomorphic to the normalization of the Koll\'ar, Shepherd-Barron, and Alexeev compactification parametrizing pairs $\left(S,\left(\frac{1}{9}+\epsilon\right)D\right)$, with $D$ the sum of the $27$ marked lines on $S$, and their stable degenerations. In the current paper, we show that the normalization assumption is not necessary as we prove that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is done by studying the automorphisms and the $\mathbb{Q}$-Gorenstein obstructions of the stable pairs parametrized by it.