Bernstein-Sato polynomial of plane curves and Yano's conjecture

Abstract

The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano in 1982 about the generic b-exponents of an irreducible plane curve. In a part of the thesis, we study the Bernstein-Sato polynomial using the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function of plane curves. Using these results we obtain a proof of Yano’s conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. This asymptotic expansion of the periods can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano’s conjecture. In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of an irreducible plane curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number in an equisingularity class of an irreducible plane curve in terms of the multiplicities of the strict transform along the minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question. This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface.