Bernstein – Sato polynomials on normal toric varieties. (English) Zbl 1402.14026

Hsiao, Jen-Chieh; Matusevich, Laura Felicia

Mich. Math. J. 67, No. 1, 117-132 (2018).

Normally, multiplier ideals and \(b\)-functions are related to some divisor (or ideals) in a smooth ambient space. This paper generalizes \(b\)-functions to the case of the ideals on normal toric varieties, and showed that this \(b\)-function is the same as the usual \(b\)-function of some ideal on a smooth variety (i.e. locally an ideal in the polynomial ring of some affine space). Moreover, since multiplier ideals have been generalized to the case of ideals on normal varieties [T. de Fernex and C. D. Hacon, Compos. Math. 145, No. 2, 393–414 (2009; Zbl 1179.14003)], the authors can get a generalization of the classical result about the roots of \(b\)-function and the log-canonical threshold [N. Budur et al., Compos. Math. 142, No. 3, 779–797 (2006; Zbl 1112.32014)], which in the simplest case, for any divisor \((f=0)\) in \(\mathbb{A}^n\), the jumping number, the log canonical threshold \(\alpha_f\) is the smallest root of \(b_f(-s)\), and any jumping number in \([\alpha_f, \alpha_f+1)\) are roots of \(b_f(-s)\).

Reviewer: Mingyi Zhang (Chicago)

MSC:

14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14F18	Multiplier ideals
14B05	Singularities in algebraic geometry

Keywords:

Bernstein-Sato polynomial; b-function; toric varieties; multiplier ideal; normal varieties

Citations:

Zbl 1179.14003; Zbl 1112.32014

References:

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