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arXiv · 2312.09774

A note on the semistability of singular projective hypersurfaces

Thomas Mordant
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Abstract

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 $δ$ of its singularities, and the dimension $s$ of its singular locus. For instance, we show that $H$ is semistable when $d \geq δ\min (N+1, s+3)$. 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 $λ$ of $\mathrm{SL}_{N+1, k}$, in terms of the numerical data in the Hilbert-Mumford criterion applied to $λ$ and to an equation $F_H$ of $H$.

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Thomas Mordant. 2024-02-07. A note on the semistability of singular projective hypersurfaces. https://doi.org/10.1007/s00209-024-03472-1

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