Low degree subvarieties of universal hypersurfaces

We study irreducible subvarieties of the universal hypersurface $\mathcal{X}\mathrm{/}B$ 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 ${\text{Sym}}^d{X}_{k(B)}$, and in particular are not collinear.