Uniformity of the minimal exponent on the GIT boundary of projective hypersurfaces
Abstract
Let X be a non-stable (in the sense of GIT) hypersurface of degree d ≥ 3 in the projective space ℙn. S. G. Park proved that the global minimal exponent of X satisfies the inequality α̃(X) ≤ (n+1)/d. In this paper, we establish the uniform sharpness of this inequality: we prove that α̃(X) = (n+1)/d holds for a general member X of every irreducible component of the strictly semistable locus. Moreover, a general point of each irreducible component of the GIT boundary ∂ MGIT admits a strictly semistable zero-weight representative with this minimal exponent. By the lower semicontinuity of the minimal exponent on the affine cone and a theorem of Mustaţă--Popa for general linear combinations of monomials, the problem is reduced to computing the log canonical thresholds, at the cone vertex, of the monomial ideals associated with the maximal strictly semistable supports with respect to a maximal torus and their weight-zero parts. Since the barycenter of the degree-d simplex lies in the convex hull of each of these supports, Howald's formula forces these thresholds to be uniformly equal to (n+1)/d.
Related
- Conformally flat free boundary minimal hypersurfaces in the ball
- Free boundary minimal hypersurfaces of Morse index one in Eucildean half-space
- Regularity and compactness theory of stable free-boundary minimal hypersurfaces
- GKZ Systems and Gauss-Manin Connections of Families of Projective Hypersurfaces
- Algebraic hyperbolicity of very general hypersurfaces in projective spaces
BibTeX
@misc{shibata2026uniformity,
title = {{Uniformity of the minimal exponent on the GIT boundary of projective hypersurfaces}},
author = {Yasutaka Shibata},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30985},
eprint = {2610.30985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30985},
}
RIS
TY - UNPB
TI - Uniformity of the minimal exponent on the GIT boundary of projective hypersurfaces
AU - Yasutaka Shibata
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30985
UR - https://arxiv.org/abs/2610.30985
AB - Let X be a non-stable (in the sense of GIT) hypersurface of degree d ≥ 3 in the projective space ℙ^n. S. G. Park proved that the global minimal exponent of X satisfies the inequality α̃(X) ≤ (n+1)/d. In this paper, we establish the uniform sharpness of this inequality: we prove that α̃(X) = (n+1)/d holds for a general member X of every irreducible component of the strictly semistable locus. Moreover, a general point of each irreducible component of the GIT boundary ∂ M^GIT admits a strictly semistable zero-weight representative with this minimal exponent. By the lower semicontinuity of the minimal exponent on the affine cone and a theorem of Mustaţă--Popa for general linear combinations of monomials, the problem is reduced to computing the log canonical thresholds, at the cone vertex, of the monomial ideals associated with the maximal strictly semistable supports with respect to a maximal torus and their weight-zero parts. Since the barycenter of the degree-d simplex lies in the convex hull of each of these supports, Howald's formula forces these thresholds to be uniformly equal to (n+1)/d.
ER -