Low degree subvarieties of universal hypersurfaces
Yifeng Huang, Borys Kadets, Olivier Martin
TL;DR
The paper proves that for large degree $d$, every degree $kd$ cycle on the universal degree $d$ hypersurface $X_{n,d}$ dominating the parameter base $B_{n,d}$ arises as the transverse intersection with a degree $k$ subvariety not contained in $X_{n,d}$. The authors combine a Grassmannian technique of Riedl–Yang, the Mumford–Roitman theorem on zero-cycles, and Cayley–Bacharach conditions under a Galois action to bootstrap from zero-cycles to higher-dimensional cycles, with an explicit degree bound $d\ge 4kn+3k^3-k^2+1$. The argument proceeds by first establishing CB-type control for zero-cycles on very general hypersurfaces, then propagating this control to full Galois orbits to produce degree $k$ curves cutting out the given cycles, and finally lifting the zero-cycle result to higher dimensions via Chiantini–Ciliberto. The work also discusses the necessity of large degree and provides examples showing density phenomena and open questions for universal complete intersections and higher codimension. This advances understanding of when low-degree subvarieties suffice to generate cycles on universal hypersurfaces and connects to Bombieri–Lang-type expectations for high-degree hypersurfaces.
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.