Low degree subvarieties of universal hypersurfaces

Yifeng Huang; Borys Kadets; Olivier Martin

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arXiv · 2506.08848

Low degree subvarieties of universal hypersurfaces

Yifeng Huang·Borys Kadets·Olivier Martin

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Abstract

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.

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Yifeng Huang, Borys Kadets, Olivier Martin. 2025-06-10. Low degree subvarieties of universal hypersurfaces. https://arxiv.org/abs/2506.08848

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