Boundary of the moduli space of stable cubic fivefolds
Yasutaka Shibata
TL;DR
This work constructs a GIT compactification of cubic fivefold moduli by adjoining strictly semistable hypersurfaces and analyzes the boundary in depth. Using a convex-geometric framework for Hilbert–Mumford weights, Luna’s centralizer reduction, and CF criteria, it identifies 21 SL(7)-distinct boundary components, each with an explicit normal form and component dimension. A detailed study of the boundary reveals two wild boundary singularities of type QH(3)_{19} and a variety of positive-dimensional boundary strata including lines, conics, quadric surfaces, CI(2,2) quartics, and quadric 3-folds. The paper also provides a Kirwan-wall-crossing description of adjacencies among boundary components, and an algorithmic non-inclusion certification using Gröbner basis and Rabinowitsch trick. Overall, dimension five marks a qualitative shift from ADE/unimodal phenomena in lower dimensions, with boundary geometry governed by a richer spectrum of singularities and adjacencies.
Abstract
Using geometric invariant theory (GIT), we construct a compactification of the moduli space of stable cubic fivefolds by adjoining strictly semistable hypersurfaces. We show that the strictly semistable locus decomposes into 21 irreducible components, and for each we give an explicit closed SL(7)-orbit representative in normal form together with its component dimension. By computing saturated Jacobian ideals of these representatives, we obtain a detailed description of boundary singularities. In contrast with cubic threefolds and fourfolds, dimension five exhibits new phenomena: exactly two closed-orbit representatives carry an isolated quasi-homogeneous hypersurface singularity of type QH(3)_{19} (corank 3 with Milnor and Tjurina numbers $μ=τ=19$), while among positive-dimensional singular loci we encounter, besides lines, smooth conics, quadric surfaces (including a rank-3 cone) and CI(2,2) space quartics, also a quadric threefold and a quadric threefold cone. We further determine the adjacency relations among boundary components as wall crossings in Kirwan's stratification, finding exactly eight nontrivial pairwise intersections. Our methods combine a convex-geometric enumeration of maximal T-strictly semistable supports for Hilbert-Mumford weights, 1-PS limits with Luna's centralizer reduction, polystability via the convex-hull and Casimiro-Florentino criteria, and Groebner-basis computations (with a Rabinowitsch trick) certifying non-inclusions.