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

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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