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

Let $X$ be a non-stable (in the sense of GIT) hypersurface of degree $d\ge 3$ in the projective space ${\mathbb{P}}^n$. S. G. Park proved that the global minimal exponent of $X$ satisfies the inequality $\tilde{\alpha }(X)\le \frac{n+1}{d}$. In this paper, we establish the uniform sharpness of this inequality: we prove that $\tilde{\alpha }(X)=\frac{n+1}{d}$ 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 $\mathrm{\partial }{M}^{\mathrm{G}\mathrm{I}\mathrm{T}}$ 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 $\frac{n+1}{d}$.