Boundary of the Moduli Space of Stable Cubic Fivefolds

This paper studies the Geometric Invariant Theory compactification of the moduli space of cubic fivefolds, namely cubic hypersurfaces X ⊂ P⁶, under the natural action of SL(7) on P(Sym³ℂ⁷). The main goal is to describe the strictly semistable boundary explicitly: its irreducible components, closed-orbit normal forms, singular loci, minimal exponents, and wall adjacency relations.

Main results

• Theorem A. The 21 families Φ1, ..., Φ21 are pairwise distinct and give the 21 irreducible components of the strictly semistable locus.
• Theorem B. Closed-orbit representatives are obtained uniformly from centralizer reduction and can be normalized explicitly.
• Theorem C. The singular loci of the 21 closed-orbit representatives are classified; isolated points occur exactly for k ∈ {1, 5, 7, 8, 9, 11, 16, 19, 21}.
• Theorem D. For each k = 1, ..., 21, the global minimal exponent of the general closed-orbit cubic fivefold in Φk equals 7/3.
• Theorem E. The codimension-one wall-adjacency graph among the 21 closed strata has 21 vertices and 56 edges, with no isolated vertices.

Singularities on the boundary

Section 5 computes Sing(Xk) for each closed-orbit representative Xk = V(φknf). The top-dimensional part of the singular locus is always very low degree: lines, smooth conics, complete intersections of two quadrics, or an elliptic quartic.

Isolated boundary singularities fall into exactly six analytic types, recorded in Table 4: E19, E21, E18, O6, Ccub, and E20. These are all quasi-homogeneous, and each has local minimal exponent 7/3. The paper names them extremal cubic fivefold singularities.

In particular, the isolated points that appear on the boundary are not accidental exceptional cases; they sit exactly at the sharp threshold in Park's stability criterion.

Adjacency and wall-crossing

Section 7 studies codimension-one wall adjacency via Kirwan wall-crossing. The adjacency relation is computed by comparing maximal slices inside the hyperplanes I(rk)=0 and matching them against the support data of the 21 families.

The final adjacency graph is substantial rather than sparse: it has 56 edges and no isolated vertices. Table 6 lists the neighbor set for each boundary component, and Figure 2 visualizes the slice-matching graph.

Methods

• Section 2. Maximal torus-strictly-semistable supports are enumerated algorithmically from the exponent simplex.
• Section 3. Passing from the maximal torus to the full SL(7)-action produces 21 inequivalent boundary families.
• Section 4. Closed-orbit normal forms are obtained using 1-PS limits and centralizer reduction.
• Section 5. Singular loci are computed from saturated Jacobian ideals.
• Section 6. Minimal exponents are computed for both isolated and positive-dimensional singular loci.
• Section 8. Non-inclusion among the 21 families is proved by six filters: dimension, apolar Betti numbers, singular-locus Hilbert functions, singular-locus degrees, minimal exponents with limits of 1-cycles, and generic stabilizer tori.

Where to look in the PDF

• Abstract and introduction: pages 1-8.
• Closed-orbit normal forms and dimensions: Section 4 and Table 2.
• Singular-locus classification: Section 5, Table 3, and Table 4.
• Minimal exponents: Section 6.
• Adjacency graph: Section 7, Table 6, and Figure 2.
• Non-inclusion proof by filters: Section 8.
• Future directions and Conjecture 9.1: Section 9.

Scripts and acknowledgments

The paper states that the computational scripts used in the project are publicly available at [Shi26], and specifically mentions scripts for Sections 2, 5, 7, and 8.

The manuscript also includes a short note on the use of AI tools, explaining that ChatGPT and Gemini were used as auxiliary tools for organizing computations, supporting code drafting/debugging, and assisting with exposition and English editing, while all mathematical results were checked by the author.