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arXiv · 2109.06303

On the degree of algebraic cycles on hypersurfaces

Matthias Paulsen
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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.

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Matthias Paulsen. 2021-09-13. On the degree of algebraic cycles on hypersurfaces. https://doi.org/10.1515/crelle-2022-0036

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