On the degree of algebraic cycles on hypersurfaces

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-09-13 DOI:10.1515/crelle-2022-0036

Matthias Paulsen

引用次数: 2

Abstract

Abstract Let X ⊂ ℙ 4 {X\subset\mathbb{P}^{4}} be a very general hypersurface of degree d ≥ 6 {d\geq 6} . Griffiths and Harris conjectured in 1985 that the degree of every curve C ⊂ X {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 {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 ℚ {\mathbb{Q}} that satisfy the conjecture.

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超曲面上代数环的度

摘要设X∧²4 {X\subset\mathbb{P} ^{4}}是一个阶数为d≥6d的非常一般的超曲面{\geq 6}。Griffiths和Harris在1985年推测，每条曲线C∧X C {\subset X的度数}都{可以被d整除。尽管在1991年Kollár取得了实质性进展，但这个猜想并没有一个单一的d值。在Kollár的方法的基础上，我们证明了这个猜想有无限多个d，最小的一个是d=5005} d=5005。这些d度集合的密度是正的。我们也证明了这一结果的高维类比，并构造了定义在π (0) {\mathbb{Q}}上的光滑超曲面来满足这个猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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