Guillem Blanco defended his PhD thesis Bernstein-Sato polynomial of plane curves and Yano’s conjectures (pdf ) on April 16th, 2020. The thesis was produced within the UPC doctoral program in Applied Mathematics and was supervised by Professors Maria Alberich-Carramiñana and Josep Álvarez Montaner Currently, he is an FWO postdoctoral fellow at the Algebra Section of KU Leuven.
Thesis summary
The research in this thesis focuses on the study of invariants of algebraic singularities. The main problem considered in the thesis was the study of the roots of the Bernstein-Sato polynomial in the case of plane curve singularities.
The roots of the Bernstein-Sato polynomial bf (s) associated to a singular polynomial f ∈ C[x1, . . . , xn] are negative rational numbers connected with many other invariants of the singularity. The main difference between the majority of the invariants of f studied in the literature and the Bernstein-Sato polynomial bf (s) is that the later is not a topological invariant. Usually, this makes bf (s) a harder object to study.
The specific problem that was solved in the thesis regarding this invariant is the determination of all the roots of bf (s) in the case of generic plane curves in the same topological class. This problem is, precisely, the statement of a long-standing conjecture posed by T. Yano in 1982. Yano’s conjecture predicts that if f is irreducible, all the roots of the Bernstein-Sato polynomial of generic curves in a topologically trivial deformation of f can be determined by the topological data of f. In addition, Yano gives an explicit set of candidates for the set of generic roots from the numerical datum of the resolution of f.
This conjecture was only known to be true in a few particular cases. The first result obtained in this direction was a proof of the general case but under some mild hypothesis on the eigenvalues of the monodromy of f. This approach consisted of studying the analytic continuation of the complex zeta function associated with the singularity [5]. The poles of this zeta function are linked to roots of the Bernstein-Sato polynomial via the Bernstein-Sato functional equation.
The approach that lead to a complete proof of the conjecture focused on the Gauss-Manin connection of an isolated singularity.
This approach consisted of constructing the asymptotic expansion of period integrals in the Milnor fiber using resolution of singularities [1]. The roots of the Bernstein-Sato polynomial can be connected with the so-called Gauss-Manin exponents using some classical results of Malgrange and Varchenko.
In the thesis other invariants of singularities were also studied [2–4, 6, 7]. For instance, the author worked on a question of Dimca and Greuel on the quotient of the Milnor and the Tjurina algebra of plane curves. In the thesis a positive answer to this question for the case of plane irreducible curves is presented [2]. More remarkably, to solve this question, a closed formula for the minimal Tjurina number in a topological class of plane branches is given, a problem that originally went back to Zariski.
Highlighted publications
References
[1] G. Blanco, Yano’s conjecture, Invent. Math., 226 (2021), 421- 465.
[2] M. Alberich-Carramiñana, P. Almirón, G. Blanco, A. MelleHernández, The minimal Tjurina number of irreducible germs of plane curve singularities, Indiana Univ. Math. J., 70 (2021), 1211-1220.
[3] M. Alberich-Carramiñana, J. Àlvarez Montaner, G. Blanco. Monomial generators of complete planar ideals, J. Algebra Appl., 20 (2021), no. 3, 2150032.
[4] P. Almirón, G. Blanco, A note on a question of Dimca and Greuel, C. R. Math. Acad. Sci. Paris, 357 (2019), no. 2, 205–208.
[5] G. Blanco, Poles of the complex zeta function of a plane curve, Adv. Math., 350 (2019), 396–439.
[6] M. Alberich-Carramiñana, J. Àlvarez Montaner, G. Blanco, Effective computation of base points of ideals in twodimensional local rings, J. Symbolic Comput., 92 (2019), 93–109.
[7] G. Blanco, F. Dachs-Cadefau, Computing multiplier ideals in smooth surfaces, Extended Abstracts February 2016. Positivity and Valuations. Research Perspectives CRM Barcelona. Trends in Mathematics. vol. 9. Birkhäuser, Basel, 2018.