The \(W(E_6)\)-invariant birational geometry of the moduli space of marked cubic surfaces (Q6594778)

Let \(Y\) denote the moduli space of marked cubic surfaces. Let \(\overline{Y}\) denote Naruki's \(W(E_6)\)-equivariant smooth projective compactification of \(Y\) [\textit{I. Naruki}, Proc. Lond. Math. Soc. (3) 45, 1--30 (1982; Zbl 0508.14005)]. Let \(\tilde{Y}\) denote the KSBA stable pair compactification of \(Y\) constructed by \textit{P. Hacking} et al. [Invent. Math. 178, No. 1, 173--227 (2009; Zbl 1205.14012)]. In this paper, the author first describes the \(W(E_6)\)-invariant effective cones of divisors and curves on \(\overline{Y}\) and \(\tilde{Y}\). Then the author computes the \(W(E_6)\)-invariant stable base locus decomposition of \(\overline{Y}\) and describes the corresponding models. The author also computes the stable base locus decomposition in the slice of the \(W(E_6)\)-invariant effective cone of \(\tilde{Y}\) given by \(K_{\tilde{Y}} + cB + dE\), where \(B\) is the sum of the boundary divisors and \(E\) is the sum of the Eckardt divisors, and describes the birational transformations corresponding to the wall-crossings.