Several sufficient conditions for projective hypersurfaces to be GIT (semi)stable

TL;DR

This work addresses when projective hypersurfaces of degree $d$ in $\mathbb{P}^n$ are (semi-)stable under $\mathrm{PGL}(n+1)$ by presenting explicit sufficient conditions in terms of $n$, $d$, the singular locus dimension $s$, and the maximal multiplicity $\delta$ of singularities. The authors employ the Hilbert-Mumford criterion and a detailed analysis of multiplicities and tangent cones to derive two main theorems that yield concrete bounds guaranteeing (semi-)stability, including refined bounds when singularities are isolated. A Hessian-based criterion is developed for the special cases $d=3$ or $4$ with isolated multiplicities $2$, connecting the Hessian rank at singular points to stability, and they deduce sharp corollaries for $A_1$ and other ADE singularities across various $(n,d)$ regimes. The results improve understanding of the moduli of hypersurfaces and provide practical checks for GIT (semi-)stability beyond smooth cases, complementing prior work such as Mordant (2024).

Abstract

In this paper, I present some sufficient conditions for projective hypersurfaces to be GIT (semi-)stable. These conditions will be presented in terms of dimension and degree of the hypersurfaces, dimension of the singular locus and multiplicities of the singular points. When singularities of the hypersurface are isolated and all have multiplicity 2, we can judge its stability via the ranks of Hessian matrices at these singular points.

Theorems & Definitions (10)

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• proof : Proof of theorem \ref{['theorem:main1']}
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• proof : Proof of proposition \ref{['proposition:finaltrick']}