On the degree of algebraic cycles on hypersurfaces

Abstract

Let $X\subset {\mathbb{P}}^4$ be a very general hypersurface of degree $d\ge 6$. Griffiths and Harris conjectured in 1985 that the degree of every curve $C\subset X$ is divisible by $d$. Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of $d$. Building on Kollár'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.

13 pages; to appear in J. Reine Angew. Math.; comments welcome

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