Bernstein-Sato ideals

We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify some hyperplanes on which all polynomials in multivariable Bernstein ideal vanish.

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.

We show that the Bernstein-Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange's formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than -2 and the multiplicity of the root -1 is equal to the (effective) dimension of the ambient space. We also give an estimate of t...

This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map f...

We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blowing-up along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.

This article is an exposition of an elementary constructive proof of canonical resolution of singularities in characteristic zero, presented in detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant and get an algorithm for canonical desingularization by successively blowing up its maximum loci. The invariant can be described by local computations that provide equations for the centres of blowing up. We describe the origin of our approach and present the...

In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider the analogous D-module in positive characteristic, and use it to define a sequence of Bernstein-Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by ...

This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as Koszul and Lie algebra methods, homological techniques, and the Bernstein-Gelfand-Gelfand correspondence, all illustrated with concrete calculations. We also explore connections of arrangements to other areas, such as De Concini-Procesi won...

This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

In 1987, C. Sabbah proved the existence of Bernstein-Sato polynomials associated with several analytic functions. The purpose of this article is to give a more elementary and constructive proof of the result of C. Sabbah based on the notion of the analytic Gr\"obner fan of a D-module.