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