The W(E6)$W(E_6)$‐invariant birational geometry of the moduli space of marked cubic surfaces

The moduli space Y=Y(E6)$Y = Y(E_6)$ 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$Y$ was restored in the 1980s by Naruki's explicit construction of a W(E6)$W(E_6)$‐equivariant smooth projective compactification Y¯${\overline{Y}}$ of Y$Y$, and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification Y∼${\widetilde{Y}}$ of Y$Y$ as a natural sequence of blowups of Y¯${\overline{Y}}$. We describe generators for the cones of W(E6)$W(E_6)$‐invariant effective divisors and curves of both Y¯${\overline{Y}}$ and Y∼${\widetilde{Y}}$. For Naruki's compactification Y¯${\overline{Y}}$, we further obtain a complete stable base locus decomposition of the W(E6)$W(E_6)$‐invariant effective cone, and as a consequence find several new W(E6)$W(E_6)$‐equivariant birational models of Y¯${\overline{Y}}$. Furthermore, we fully describe the log minimal model program for the KSBA compactification Y∼${\widetilde{Y}}$, with respect to the divisor KY∼+cB+dE$K_{{\widetilde{Y}}} + cB + dE$, where B$B$ is the boundary and E$E$ is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.