On the irreducibility of Hessian loci of cubic hypersurfaces
We study the problem of the irreducibility of the Hessian variety ${\mathcal {H}}_f$ associated with a smooth cubic hypersurface $V(f)\subset {\mathbb {P}}^n$ . We prove that when $n\leq 5$ , ${\mathcal {H}}_f$ is normal and irreducible if and only if f is not of Thom-Sebas...
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| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425000362/type/journal_article |
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| author | Davide Bricalli Filippo Francesco Favale Gian Pietro Pirola |
| author_facet | Davide Bricalli Filippo Francesco Favale Gian Pietro Pirola |
| author_sort | Davide Bricalli |
| collection | DOAJ |
| description | We study the problem of the irreducibility of the Hessian variety
${\mathcal {H}}_f$
associated with a smooth cubic hypersurface
$V(f)\subset {\mathbb {P}}^n$
. We prove that when
$n\leq 5$
,
${\mathcal {H}}_f$
is normal and irreducible if and only if f is not of Thom-Sebastiani type (i.e., if one cannot separate its variables by changing coordinates). This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities. |
| format | Article |
| id | doaj-art-acaf2f1935e04ecc9c50c8bfe096cf0d |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-acaf2f1935e04ecc9c50c8bfe096cf0d2025-08-20T03:53:43ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.36On the irreducibility of Hessian loci of cubic hypersurfacesDavide Bricalli0https://orcid.org/0009-0009-8109-9551Filippo Francesco Favale1https://orcid.org/0000-0002-2924-184XGian Pietro Pirola2https://orcid.org/0000-0002-2007-2525Università degli Studi di Pavia, Dipartimento di Matematica, Via Adolfo Ferrata 5, 27100 Pavia, Italy; E-mail: INdAM (GNSAGA); E-mail:Università degli Studi di Pavia, Dipartimento di Matematica, Via Adolfo Ferrata 5, 27100 Pavia, Italy; E-mail:Università degli Studi di Pavia, Dipartimento di Matematica, Via Adolfo Ferrata 5, 27100 Pavia, ItalyWe study the problem of the irreducibility of the Hessian variety ${\mathcal {H}}_f$ associated with a smooth cubic hypersurface $V(f)\subset {\mathbb {P}}^n$ . We prove that when $n\leq 5$ , ${\mathcal {H}}_f$ is normal and irreducible if and only if f is not of Thom-Sebastiani type (i.e., if one cannot separate its variables by changing coordinates). This also generalizes a result of Beniamino Segre dealing with the case of cubic surfaces. The geometric approach is based on the study of the singular locus of the Hessian variety and on infinitesimal computations arising from a particular description of these singularities.https://www.cambridge.org/core/product/identifier/S2050509425000362/type/journal_article14J7014M1214J1714J3014J35 |
| spellingShingle | Davide Bricalli Filippo Francesco Favale Gian Pietro Pirola On the irreducibility of Hessian loci of cubic hypersurfaces Forum of Mathematics, Sigma 14J70 14M12 14J17 14J30 14J35 |
| title | On the irreducibility of Hessian loci of cubic hypersurfaces |
| title_full | On the irreducibility of Hessian loci of cubic hypersurfaces |
| title_fullStr | On the irreducibility of Hessian loci of cubic hypersurfaces |
| title_full_unstemmed | On the irreducibility of Hessian loci of cubic hypersurfaces |
| title_short | On the irreducibility of Hessian loci of cubic hypersurfaces |
| title_sort | on the irreducibility of hessian loci of cubic hypersurfaces |
| topic | 14J70 14M12 14J17 14J30 14J35 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425000362/type/journal_article |
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