The GIT Boundary of Quintic Threefolds

Overview

This paper determines the strictly semistable boundary in the Geometric Invariant Theory (GIT) quotient for quintic threefolds, that is, degree-five hypersurfaces X ⊂ ℙ⁴. The group G = SL(5) acts naturally on ℙ(W), where W = Sym⁵ ℂ⁵. Writing M^GIT = ℙ(W)^ss // G and π for the quotient map, the boundary studied here is π(ℙ(W)^ss ∖ ℙ(W)^s). It is not the entire complement of the smooth-quintic locus in the GIT quotient.

∂M^GIT = Θ₁ ∪ ⋯ ∪ Θ₂₁.

Source: Version 2, Theorem A (p. 2).

Throughout this overview, “general” means on the nonempty Zariski-open parameter loci specified in the paper. The normal-form, stabilizer, and singularity assertions are not claims about every parameter specialization; the displayed normal-form parameterizations are not asserted to be unique.

The 21 quotient-side boundary components

Why 38 supports give 21 components

The Hilbert–Mumford calculation enumerates 38 maximal strictly semistable monomial supports for the diagonal maximal torus, up to coordinate permutation. Their G-saturations cover the strictly semistable locus, but this support-indexed cover is not yet an irreducible-component decomposition of the quotient.

Zero-weight reduction shows that reversing a supporting half-space does not change its quotient image: Φ(r) = Φ(−r). Reversal pairs 34 of the support labels into 17 pairs and fixes four labels, namely 13, 14, 19, and 36. Thus 38 = 17 × 2 + 4 oriented supports yield 17 + 4 = 21 quotient families.

Sources: Section 4: zero-weight reduction and supporting-hyperplane invariance; Section 6: the component theorem.

Component catalog

In the following table, Φ_k denotes a support-indexed quotient family, k(a) is the chosen normal-form case, and the primitive normal r_a defines the generic connected projective stabilizer H_a = λ_ra(G_m). The last column is the quotient dimension.

Source: Version 2, Table 2 (p. 5). The component index a and the support-case label k(a) differ from component 16 onward.

The four largest components are Θ₂₁ (dimension 24), Θ₁₈ (19), Θ₂₀ (18), and Θ₁₉ (15). The comparison in Section 7 identifies these with Lakhani’s four first-level families. The other 17 are additional irreducible components, not contained in their union. See Section 7.

Closed orbits and the non-inclusion argument

Weight-zero limits and Luna’s centralizer reduction produce explicit polystable normal forms. Exact rank certificates then show that the generic connected projective stabilizers are precisely the one-dimensional tori listed above. Their pairwise nonconjugacy, together with Luna’s étale slice theorem, rules out containments between distinct quotient families. The component theorem therefore uses stabilizers, not merely a comparison of dimensions.

Sources: Section 5, Section 6, and Appendix A. The 21 normal-form equations and their genericity conditions are collected in Table 1 (pp. 3–4).

Singularities and the minimal exponent

For a general closed-orbit representative in each component, the paper computes the saturated Jacobian scheme and its local analytic stratification. Positive-dimensional singular supports include configurations of lines, smooth conics, cuspidal plane curves, a plane, and a smooth quartic surface. The catalog distinguishes generic transverse structures from special strata where the transverse model changes, and records all isolated singular points.

Source: Version 2, Table 3 (pp. 6–7); the component-by-component calculations are in Appendix B.

The 11 isolated weighted-homogeneous local families

The table below summarizes the integral weights, Milnor numbers, Yonemura row numbers, and occurrences of the 11 families. Here P₀ = (1 : 0 : 0 : 0 : 0) and P_∞ = (0 : 0 : 0 : 0 : 1), in the coordinates of the normal forms. The defining local equations and parameter conditions are given in Table 4 and Definitions 8.1–8.11 of the paper.

Source: Version 2, Table 4 (pp. 7–8).

“Yonemura-type” is a statement about normalized weights. Each normalized weight system occurs in Yonemura’s Table 2.2. This does not assert that every member of one of these local families is analytically equivalent to the particular polynomial displayed in that list. The occurrence-and-exhaustion statement concerns general closed-orbit representatives on the quintic GIT boundary. See the explanation following Table 4.

The extremal value

Every isolated family above has w₁ + w₂ + w₃ + w₄ = D, so its local minimal exponent is one. The local analysis along the positive-dimensional singular loci shows that no stratum forces a value below one, while at least one stratum in every component attains one. Consequently, for a general X_a = V(F_a),

$\tilde{\alpha }$(X_a) = 1 = (4 + 1)/5, a = 1, …, 21.

“Extremal” refers to equality with the critical value (n + 1)/d for degree-five hypersurfaces in ℙ⁴.

Sources: Introduction (pp. 10–11) and Appendix B.

Guide to Version 2

All references below point to the corresponding pages of the Version 2 PDF.