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A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2020-12-01 , DOI: 10.1016/j.jpaa.2020.106435
Eloísa Grifo

Abstract The symbolic powers I ( n ) of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers I n , but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I ( n ) ⊆ I m holds. When I is an ideal of height 2 in a regular ring, I ( 3 ) ⊆ I 2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curves ( t a , t b , t c ) . More generally, given a radical ideal I of big height h, while the containment I ( h n − h + 1 ) ⊆ I n conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee I ( h n − h + 1 ) ⊆ I n for n ≫ 0 .

中文翻译:

Harbourne 猜想的稳定版本和空间单项式曲线的包含问题

摘要 多项式环中激进理想 I 的符号幂 I ( n ) 由在 I 定义的变体中消失到 n 阶的函数组成。这些函数不一定与普通的代数幂 I n 重合,但它是很自然地比较这两个概念。包含问题包括确定满足 I ( n ) ⊆ I m 的 n 和 m 的值。当 I 是规则环中高度为 2 的理想时, I ( 3 ) ⊆ I 2 可能会失败,但我们证明了这种包容性确实适用于空间单项式曲线 ( ta , tb , tc ) 的定义理想。更一般地,给定一个大高度 h 的激进理想 I,而 Harbourne 推测的包含 I ( hn − h + 1 ) ⊆ I n 不一定对所有 n 都成立,我们给出了充分条件来保证 I ( hn − h + 1 ) ⊆ I n 对于 n ≫ 0 。
更新日期:2020-12-01
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