Low-degree subvarieties of universal hypersurfaces

We study irreducible subvarieties of the universal hypersurface 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 D 3, rational points are dense in Symd Xk.B/, and in particular are not collinear.