The W ( E 6 ) $W(E_6)$ -invariant birational geometry of the moduli space of marked cubic surfaces

Abstract

The moduli space $Y=Y\left(E_6\right)$ 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 $W\left(E_6\right)$-equivariant smooth projective compactification $\overline{Y}$ of $Y$, and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd-Barron–Alexeev (KSBA) stable pair compactification $\stackrel{\sim }{Y}$ of $Y$ as a natural sequence of blowups of $\overline{Y}$. We describe generators for the cones of $W\left(E_6\right)$-invariant effective divisors and curves of both $\overline{Y}$ and $\stackrel{\sim }{Y}$. For Naruki's compactification $\overline{Y}$, we further obtain a complete stable base locus decomposition of the $W\left(E_6\right)$-invariant effective cone, and as a consequence find several new $W\left(E_6\right)$-equivariant birational models of $\overline{Y}$. Furthermore, we fully describe the log minimal model program for the KSBA compactification $\stackrel{\sim }{Y}$, with respect to the divisor $K_{\stackrel{\sim }{Y}}+cB+dE$, where $B$ is the boundary and $E$ is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.