The Hessian correspondence of hypersurfaces of degree 3 and 4
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914625370456064 |
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| author | Sendra-Arranz, Javier |
| author_facet | Sendra-Arranz, Javier |
| contents | Let $X$ be a hypersurface, of degree $d$, in an $n$--dimensional projective space. The Hessian map is a rational map from $X$ to the projective space of symmetric matrices that sends a point $p\in X$ to the Hessian matrix of the defining polynomial of $X$ evaluated at $p$. The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via the Hessian map. In this paper, we study this correspondence for hypersurfaces with Waring rank at most $n+1$ and for hypersurfaces of degree $3$ and $4$. We prove that, for hypersurfaces with Waring rank $k\leq n+1$, the map is birational onto its image for $d$ even, and it is generically finite of degree $2^{k-1}$ for $d$ odd. We prove that, for degree $3$ and $n=1$, the map is two to one, and that, for degree $3$ and $n\geq 2$, and for degree $4$, the Hessian correspondence is birational. In this study, we introduce the $k$--gradients varieties and analyze their main properties. We provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10415 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Hessian correspondence of hypersurfaces of degree 3 and 4 Sendra-Arranz, Javier Algebraic Geometry 14J70, 14E05, 14Q20, 14N05 Let $X$ be a hypersurface, of degree $d$, in an $n$--dimensional projective space. The Hessian map is a rational map from $X$ to the projective space of symmetric matrices that sends a point $p\in X$ to the Hessian matrix of the defining polynomial of $X$ evaluated at $p$. The Hessian correspondence is the map that sends a hypersurface to its Hessian variety; i.e. the Zariski closure of its image via the Hessian map. In this paper, we study this correspondence for hypersurfaces with Waring rank at most $n+1$ and for hypersurfaces of degree $3$ and $4$. We prove that, for hypersurfaces with Waring rank $k\leq n+1$, the map is birational onto its image for $d$ even, and it is generically finite of degree $2^{k-1}$ for $d$ odd. We prove that, for degree $3$ and $n=1$, the map is two to one, and that, for degree $3$ and $n\geq 2$, and for degree $4$, the Hessian correspondence is birational. In this study, we introduce the $k$--gradients varieties and analyze their main properties. We provide effective algorithms for recovering a hypersurface from its Hessian variety, for degree $3$ and $n\geq 1$, and for degree $4$ and $n$ even. |
| title | The Hessian correspondence of hypersurfaces of degree 3 and 4 |
| topic | Algebraic Geometry 14J70, 14E05, 14Q20, 14N05 |
| url | https://arxiv.org/abs/2307.10415 |