arXiv subjects
Publications and source records attributed to Guillem Blanco.
For an isolated complete intersection singularity (ICIS), we define and study its Gauss-Manin system and its associated Hodge filtration. We show the relation between the Hodge filtration and a generalized Brieskorn lattice and study conditions for the existence of a microlocal structure. Using these ideas, we relate the residue on the saturated Brieskorn lattice with a $b$-function introduced by Torrelli.
We study a set of topological roots of the local Bernstein-Sato polynomial of arbitrary plane curve singularities. These roots are characterized in terms of certain divisorial valuations and the numerical data of the minimal log resolution. In particular, this set of roots strictly contains both the opposites of the jumping numbers in $(0, 1)$ and the poles of the motivic zeta function counted with multiplicity. As a consequence, we prove the multiplicity part of the Strong Monodromy Conjecture for $n = 2$.
We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of b-functions for isolated quasihomogeneous hypersurfaces, and more generally for semi-quasihomogeneous hypersurfaces. We also give a strange generalization allowing a twist by certain differential forms.
We present a proof of a conjecture proposed by T. Yano about the generic $b$-exponents of irreducible plane curve singularities.
We present an algorithm to compute the Hodge ideals of $\mathbb{Q}$-divisors associated to any reduced effective divisor $D$. The computation of the Hodge ideals is based on an algorithm to compute parts of the $V$-filtration of Malgrange and Kashiwara on $\iota_{+}\mathscr{O}_X(*D)$ and the characterization of the Hodge ideals in terms of this $V$-filtration. In particular, this gives a new algorithm to compute the multiplier ideals and the jumping numbers of any effective divisor.
In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer, Wall and Mattei.
In this note we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and Tjurina numbers of an isolated plane curve singularity in the cases of one Puiseux pair and semi-quasi-homogeneous singularities.
We study the poles and residues of the complex zeta function $ f^s $ of a plane curve. We prove that most non-rupture divisors do not contribute to poles of $ f^s $ or roots of the Bernstein-Sato polynomial $ b_f(s) $ of $ f $. For plane branches we give an optimal set of candidates for the poles of $ f^s $ from the rupture divisors and the characteristic sequence of $ f $. We prove that for generic plane branches $ f_{gen} $ all the candidates are poles of $ f_{gen}^s $. As a consequence, we prove Yano's conjecture for any number of characteristic exponents if the eigenvalues of the monodromy of $ f $ are different.
We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals.
In this work we describe a minimal log-resolution of an ideal in a smooth complex surface from the minimal log-resolution of its generators.