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cubicfivefold.pdf

Boundary of the Moduli Space of Stable Cubic Fivefolds

Author: Yasutaka Shibata

This paper constructs the geometric invariant–theoretic (GIT) compactification of the moduli space of cubic fivefolds—degree‑3 hypersurfaces in projective space P6—by adjoining the strictly semistable locus as a boundary. A central outcome is a complete description of that boundary: it decomposes into 21 irreducible components, each with an explicit closed‑orbit representative (normal form), with dimensions recorded component‑by‑component (see Table 2, p. 48). The work also shows that dimension five marks a qualitative threshold: unlike cubic threefolds and fourfolds, wild isolated hypersurface singularities appear on the boundary of cubic fivefolds.

At a glance

  • Main classification. The strictly semistable locus splits into 21 irreducible components, and for each component the paper provides a closed‑orbit representative in normal form together with its dimension (see Table 2, p. 48).
  • Singularity profile on the boundary. Most boundary models have positive‑dimensional singular loci—lines, smooth conics, quadric surfaces (including a rank‑3 cone), non‑degenerate space quartics CI(2,2), and linear spaces P2 or P3. Strikingly, exactly two closed‑orbit representatives (Cases k=1 and k=6) have an isolated quasi‑homogeneous hypersurface singularity of type QH(3)19 with Milnor and Tjurina numbers μ = τ = 19 (see Theorem C and Table 3, pp. 54–55).
  • Adjacency (wall‑crossing). The paper determines all pairwise adjacencies among boundary components as wall‑crossings in Kirwan’s stratification; there are eight non‑empty intersections (see Theorem 6.1).
  • Scale of the moduli problem. The GIT moduli space of cubic fivefolds has dimension 35 (number of degree‑3 monomials in seven variables minus dim(PGL7)).

Methods in brief

  • A convex‑geometric analysis of Hilbert–Mumford weights on W = Sym37 (with a fixed maximal torus) enumerates maximal torus‑strictly‑semistable supports. An algorithmic search (Algorithm 2.2) yields 22 torus families, one unstable; modulo SL(7) this produces the desired 21 families (Theorem 3.6).
  • Closed‑orbit representatives are obtained via 1‑parameter limits and Luna’s centralizer reduction; polystability is certified either by the convex‑hull criterion (for toric centralizers) or by the Casimiro–Florentino symmetric‑1‑PS criterion (for non‑toric centralizers), producing explicit normal forms summarized in Table 2 (Section 4).
  • Singular loci are computed from the saturated Jacobian ideal of each normal form, giving precise set‑theoretic descriptions and, when present, local invariants at isolated points (Section 5; Table 3). Gröbner‑basis computations (Macaulay2, Singular) certify both these singularity statements and the non‑inclusion relations among distinct families (Sections 5 and 7).

Why this matters

The work clarifies the geometric boundary of the moduli space of cubic fivefolds with a fully explicit, computation‑backed classification. It shows that in dimension five the boundary leaves the ADE/unimodal world familiar from threefolds and fourfolds, exhibiting genuinely wild isolated singularities. The normal forms, component dimensions, and wall‑crossing adjacencies provide a concrete toolkit for studying degenerations of higher‑dimensional cubic hypersurfaces and for connecting GIT stability with singularity theory.

Further details

  • Normal forms & component dimensions: see Table 2 (page 48).
  • Singularities summary: see Table 3 (pages 54–55).
  • Adjacency list: see Theorem 6.1 (Section 6).
  • Computation scripts: the paper points to a public archive with all scripts used for Sections 2, 5, 6, and 7.