2023/09/26 by Nolan Schock, Schock, Nolan
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2309.15264
openalex publication_date 2023/09/26 · openalex created_date 2023/09/30 · openalex updated_date 2026/07/28
The moduli space Y = Y(E6) 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(E6)-equivariant smooth projective compactification Y of Y, and in the 2000s by Hacking, Keel, and Tevelev's construction of the KSBA stable pair compactification \widetildeY of Y as a natural sequence of blowups of Y. We describe generators for the cones of W(E6)-invariant effective divisors and curves of both Y and \widetildeY. For Naruki's compactification Y, we further obtain a complete stable base locus decomposition of the W(E6)-invariant effective cone, and as a consequence find several new W(E6)-equivariant birational models of Y. Furthermore, we fully describe the log minimal model program for the KSBA compactification \widetildeY, with respect to the divisor K_\widetildeY + cB + dE, where B is the boundary and E is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.