超曲面上代数环的度

IF 1.2 1区数学 Q1 MATHEMATICS

Journal fur die Reine und Angewandte Mathematik Pub Date : 2021-09-13 DOI:10.1515/crelle-2022-0036

Matthias Paulsen

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引用次数: 2

摘要

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。

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On the degree of algebraic cycles on hypersurfaces

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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来源期刊

Journal fur die Reine und Angewandte Mathematik 数学-数学

CiteScore

2.50

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle"s Journal. In the almost 180 years of its existence, Crelle"s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI"s Journal Citation Report.