A note on the semistability of singular projective hypersurfaces

In this note, we give sufficient conditions for the (semi)stability of a hypersurface H of \(\mathbb {P}^N_k\) ${\mathbb{P}}_k^N$ in terms of its degree d, the maximal multiplicity \(\delta \) $\delta$ of its singularities, and the dimension s of its singular locus. For instance, we show that H is semistable when \(d \ge \delta \min (N+1, s+3)\) $d\ge \delta min(N+1,s+3)$ . The proof relies in particular on Benoist’s lower bound for the dimension of the intersection of the singular locus \(H_{\textrm{sing}}\) $H_{\text{sing}}$ of H with some linear subspace of \(\mathbb {P}^N_k\) ${\mathbb{P}}_k^N$ associated to a one-parameter subgroup \(\lambda \) $\lambda$ of \(\textrm{SL}_{N+1, k}\) ${\text{SL}}_{N+1,k}$ , in terms of the numerical data in the Hilbert–Mumford criterion applied to \(\lambda \) $\lambda$ and to an equation \(F_H\) $F_H$ of H.