Low-degree subvarieties of universal hypersurfaces

Abstract We study irreducible subvarieties of the universal hypersurface <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">X</m:mi> <m:mo>/</m:mo> <m:mi>B</m:mi> </m:mrow> </m:math> \mathcal{X}/B of degree 𝑑 and dimension 𝑛. We prove that, when 𝑑 is sufficiently large, a degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> 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–Roĭtman 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> d=3 , rational points are dense in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>Sym</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo lspace="0.167em">⁡</m:mo> <m:msub> <m:mi>X</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:math> \operatorname{Sym}^{d}X_{k(B)} , and in particular are not collinear.