Low degree subvarieties of universal hypersurfaces

TL;DR

This paper investigates the structure of subvarieties within universal hypersurfaces, proving that high-degree subvarieties are derived from intersections with lower-degree varieties, thus answering a question by Farb and Ma.

Contribution

It establishes a new geometric characterization of subvarieties in universal hypersurfaces for large degrees, using Grassmannian techniques and rational equivalence theories.

Findings

High-degree subvarieties dominate the base and are intersections with lower-degree varieties.

The large degree condition is essential; for degree 3, rational points are dense and not collinear.

The methods involve Grassmannian techniques, Mumford-Roitman theorem, and Cayley-Bacharach analysis.

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.