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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2312.09774 |
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| _version_ | 1866911880337948672 |
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| author | Mordant, Thomas |
| author_facet | Mordant, Thomas |
| contents | 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 $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09774 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the semistability of singular projective hypersurfaces Mordant, Thomas Algebraic Geometry 14L24 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 $H$. |
| title | A note on the semistability of singular projective hypersurfaces |
| topic | Algebraic Geometry 14L24 |
| url | https://arxiv.org/abs/2312.09774 |