On the degree of algebraic cycles on hypersurfaces

TL;DR

This paper proves a long-standing conjecture that the degree of any curve on a very general hypersurface in P^4 is divisible by the degree d, for infinitely many degrees d, including some with positive density.

Contribution

It extends Kollár's method to prove Griffiths and Harris's conjecture for infinitely many degrees d, including explicit examples with positive density.

Findings

Proved the conjecture for infinitely many degrees d, smallest being 5005.

Constructed smooth hypersurfaces over Q satisfying the conjecture.

Established a higher-dimensional analogue of the result.

Abstract

Let $X\subset\mathbb P^4$ be a very general hypersurface of degree $d\ge6$. Griffiths and Harris conjectured in 1985 that the degree of every curve $C\subset X$ is divisible by $d$. Despite substantial progress by Koll\'ar in 1991, this conjecture is not known for a single value of $d$. Building on Koll\'ar's method, we prove this conjecture for infinitely many $d$, the smallest one being $d=5005$. The set of these degrees $d$ has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over $\mathbb Q$ that satisfy the conjecture.