Uniformity of the minimal exponent on the GIT boundary of projective hypersurfaces

Overview

This paper studies the global minimal exponent of projective hypersurfaces in the strictly semistable locus of Geometric Invariant Theory (GIT). For a hypersurface X ⊂ ℙⁿ of degree d ≥ 3, Park's stability criterion gives the upper bound $\tilde{\alpha }$(X) ≤ (n + 1)/d whenever X is not GIT stable.

The paper proves that this bound is uniformly sharp: a general member of every irreducible component of the strictly semistable locus attains exactly the same critical value. It also constructs zero-weight representatives with this value over a nonempty open subset of each irreducible component of the GIT boundary.

The argument explains the uniform values observed in the author's studies of cubic fivefolds and quintic threefolds through a common geometric feature: the barycenter of the degree simplex lies in the convex hull of the relevant monomial supports.

Sources: Abstract and introduction (pp. 1–2); Park's criterion, Theorem 4.1 (p. 9).

Main theorem

Work over ℂ, with n ≥ 1 and d ≥ 3. Write Σ^ss for the locus of hypersurfaces that are semistable but not stable under SL(n + 1), and let π : Σ^ss → ∂M^GIT be the quotient map to the strictly semistable GIT boundary.

Zero-weight representatives on the quotient

For every irreducible component B of ∂M^GIT, there exist a nonempty open subset B^ˆ ⊆ B and a nontrivial one-parameter subgroup λ : 𝔾_m → SL(n + 1) such that each b ∈ B^ˆ admits a strictly semistable representative [f] with

Here the middle equality holds for every t ∈ 𝔾_m: the defining polynomial lies in the weight-zero subspace for this one-parameter subgroup. The more precise Theorem 4.2 proves the equality for general polynomials in both the maximal strictly torus-semistable monomial families and their weight-zero subspaces.

Sources: Main Theorem (p. 1); Theorem 4.2 and its proof (pp. 10–11).

Scope of the result

Throughout the paper, hypersurfaces are treated as effective Cartier divisors defined by their equations; reducedness is not assumed. See Section 1 (p. 3).

Idea of the proof

The proof reduces the singularity invariant to a convex-geometric computation that is identical for every support in a finite covering.

1. Cover the strictly semistable locus by monomial families

   The numerical criterion and the enumeration of maximal supports produce finitely many families associated with I(r_k)_≥0. The closures of their SL(n + 1)-saturations, intersected with the semistable locus, form a finite covering of the strictly semistable locus. Lemma 2.3 identifies the shared condition: both the support and its weight-zero part contain the barycenter of the degree simplex in their convex hulls.

   η = d/(n + 1) · (1, …, 1) ∈ Conv(I(r_k)₌₀).

   Lemma 2.3 and Theorem 2.4 (pp. 6–7).

2. Reduce the quotient image to the weight-zero subspace

   For a semistable polynomial in one of the nonnegative-weight families, write f = f₀ + f_>0. Its one-parameter limit is f₀, which remains semistable and represents the same point of the GIT quotient. The resulting zero-weight families give a finite covering of the boundary after taking closures of their quotient images.

   Lemma 3.1 and Theorem 3.2 (pp. 7–9).

3. Compute the minimal exponent at the vertex of the affine cone

   For either support S, let 𝔞_S be its monomial ideal. A theorem of Mustaţă–Popa identifies the cone-vertex minimal exponent of a general linear combination of these monomials with the log canonical threshold of this ideal. Howald's formula computes that threshold from the Newton polyhedron.

   Every generator has degree d, and the convex hull contains η. Hence the diagonal ray first enters the Newton polyhedron at η, with parameter τ = d/(n + 1).

   ${\tilde{\alpha }}_v$(f) = lct_v(𝔞_S) = 1/τ = (n + 1)/d.

   This is the threshold of a monomial ideal, so it may exceed 1. The argument does not require an isolated singularity at the cone vertex.

   Proof of Theorem 4.2, equations (3)–(4) (p. 10).

4. Match the lower bound with Park's upper bound

   Invariance under smooth pullback and lower semicontinuity transfer the cone-vertex calculation to all singular points of the projective hypersurface, giving $\tilde{\alpha }$(X) ≥ (n + 1)/d. Park's criterion then implies semistability. The support makes the hypersurface non-stable, so Park's upper bound gives the reverse inequality and therefore equality. The finite coverings carry this conclusion to every irreducible component.

   Completion of the proof of Theorem 4.2 (p. 11).

Connection with earlier research

The introduction traces the uniformity question to the author's work on cubic fivefolds and quintic threefolds. The theorem gives a conceptual explanation for the common values observed in those settings. Substituting their ambient dimensions and degrees gives:

Source: Motivation in the introduction (p. 1). The home page includes the author's note on the cubic fivefold paper.

Guide to the paper

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