Jumping coefficients of multiplier ideals
Jumping coefficients of multiplier ideals
复制标题
乘数理想的跳跃系数
DOI：
10.1215/s0012-7094-04-12333-4
复制
发表时间：
2003
影响因子：
2.5
通讯作者：
Dror Varolin
中科院分区：
文献类型：
--
作者：
L. Ein;R. Lazarsfeld;Karen E. Smith;Dror Varolin
本文研究光滑复簇上通过乘子理想附加到有效因子或理想层上的局部不变量。首先考虑（至少是隐含地）由Libgober和Loeser和Vaquie，这些跳跃系数由有理数的递增序列组成，从除数或理想的对数标准阈值开始。他们编码有趣的几何和代数信息，我们表明，他们自然出现在几个不同的背景下。给定一个只有孤立奇点的多项式f，Varchenko，Loeser和Vaquie的结果暗示：如果\xi是f = 0在区间（0，1]内的跳跃数，则-\xi是f的Bernstein-Sato多项式的根。我们适应一个参数的Kollar显示证明，这也成立时，奇异轨迹的f有积极的层面。在一个更代数的方向，我们表明，这样的跳跃系数的数量的界限的一致阿廷里斯数的主要理想（f）在Huneke的意义：在孤立的奇点的情况下，这反过来又导致涉及Milnor和Tyurina数的f的界限。沿着这条路，我们建立了一个一般性的结果乘数雅可比理想。我们还探讨了这些想法的扩展设置分级家庭的理想。这篇论文包含了许多具体的例子。
We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser and Vaquie imply that if \xi is a jumping number of f = 0 lying in the interval (0, 1], then -\xi is a root of the Bernstein-Sato polynomial of f. We adapt an argument of Kollar to show prove that this holds also when the singular locus of f has positive dimension. In a more algebraic direction, we show that the number of such jumping coefficients bounds the uniform Artin-Rees number of the principal ideal (f) in the sense of Huneke: in the case of isolated singularities, this in turn leads to bounds involving the Milnor and Tyurina numbers of f . Along the way, we establish a general result relating multiplier to Jacobian ideals. We also explore the extension of these ideas to the setting of graded families of ideals. The paper contains many concrete examples.