Veronese Avoiding Hypersurfaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915974354042880 |
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| author | Ilardi, Giovanna Nejad, Abbas Nasrollah Tafazolian, Saeed |
| author_facet | Ilardi, Giovanna Nejad, Abbas Nasrollah Tafazolian, Saeed |
| contents | We introduce Veronese-Avoiding hypersurfaces, inspired by the theory of associated forms of Alper--Isaev. In the smooth case, we reinterpret their criterion via Macaulay inverse systems: the Veronese-Avoiding condition is equivalent to the non-degeneracy of the associated form. In the singular case, our main theorem shows that a reduced hypersurface with exactly $n$ isolated singular points is Veronese-Avoiding if and only if these points are ordinary nodes in general linear position; we also classify singular plane cubics and treat fewer than $n$ nodes via a natural rational map. We then study the parameter space, proving local closedness and identifying a distinguished irreducible nodal locus. Finally, we prove a Lefschetz-type consequence for the Milnor algebra in degree $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01541 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Veronese Avoiding Hypersurfaces Ilardi, Giovanna Nejad, Abbas Nasrollah Tafazolian, Saeed Algebraic Geometry Commutative Algebra We introduce Veronese-Avoiding hypersurfaces, inspired by the theory of associated forms of Alper--Isaev. In the smooth case, we reinterpret their criterion via Macaulay inverse systems: the Veronese-Avoiding condition is equivalent to the non-degeneracy of the associated form. In the singular case, our main theorem shows that a reduced hypersurface with exactly $n$ isolated singular points is Veronese-Avoiding if and only if these points are ordinary nodes in general linear position; we also classify singular plane cubics and treat fewer than $n$ nodes via a natural rational map. We then study the parameter space, proving local closedness and identifying a distinguished irreducible nodal locus. Finally, we prove a Lefschetz-type consequence for the Milnor algebra in degree $1$. |
| title | Veronese Avoiding Hypersurfaces |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2605.01541 |