The $W(E_6)$-invariant birational geometry of the moduli space of marked cubic surfaces

TL;DR

This paper studies the birational geometry of the moduli space of marked cubic surfaces, focusing on $W(E_6)$-invariant divisors, cones, and the log minimal model program for two key compactifications.

Contribution

It provides explicit descriptions of invariant divisor and curve cones, stable base locus decompositions, and the full log minimal model program for the moduli space's compactifications.

Findings

Generated cones of invariant divisors and curves for both compactifications.

Obtained a complete stable base locus decomposition for Naruki's compactification.

Described the full log minimal model program for the KSBA compactification.

Abstract

The moduli space $Y = Y(E_6)$ of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth century work of Cayley and Salmon. Modern interest in $Y$ was restored in the 1980s by Naruki's explicit construction of a $W(E_6)$-equivariant smooth projective compactification $\overline{Y}$ of $Y$, and in the 2000s by Hacking, Keel, and Tevelev's construction of the KSBA stable pair compactification $\widetilde{Y}$ of $Y$ as a natural sequence of blowups of $\overline{Y}$. We describe generators for the cones of $W(E_6)$-invariant effective divisors and curves of both $\overline{Y}$ and $\widetilde{Y}$. For Naruki's compactification $\overline{Y}$, we further obtain a complete stable base locus decomposition of the $W(E_6)$-invariant effective cone, and as a consequence find several new $W(E_6)$-equivariant birational models of $\overline{Y}$. Furthermore, we fully describe the log minimal model program for the KSBA compactification $\widetilde{Y}$, with respect to the divisor $K_{\widetilde{Y}} + cB + dE$, where $B$ is the boundary and $E$ is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.