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

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