Low degree subvarieties of universal hypersurfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912869585518592 |
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| author | Huang, Yifeng Kadets, Borys Martin, Olivier |
| author_facet | Huang, Yifeng Kadets, Borys Martin, Olivier |
| contents | We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a family of degree $k$ projective varieties parametrized by $B$. This answers a question raised independently by Farb and Ma. Our main tools consist of a Grassmannian technique due to Riedl and Yang, a theorem of Mumford-Roitman on rational equivalence of zero-cycles, and an analysis of Cayley-Bacharach conditions in the presence of a Galois action. We also show that the large degree assumption is necessary; for $d=3$, rational points are dense in $\text{Sym}^dX_{k(B)}$, and in particular are not collinear. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low degree subvarieties of universal hypersurfaces Huang, Yifeng Kadets, Borys Martin, Olivier Algebraic Geometry Number Theory 14J70, 14J20, 14G05 We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a family of degree $k$ projective varieties parametrized by $B$. This answers a question raised independently by Farb and Ma. Our main tools consist of a Grassmannian technique due to Riedl and Yang, a theorem of Mumford-Roitman on rational equivalence of zero-cycles, and an analysis of Cayley-Bacharach conditions in the presence of a Galois action. We also show that the large degree assumption is necessary; for $d=3$, rational points are dense in $\text{Sym}^dX_{k(B)}$, and in particular are not collinear. |
| title | Low degree subvarieties of universal hypersurfaces |
| topic | Algebraic Geometry Number Theory 14J70, 14J20, 14G05 |
| url | https://arxiv.org/abs/2506.08848 |