Patricio Almirón Cuadros defended his PhD thesis Analytic invariants of isolated hypersurface singularities and combinatorial invariants of numerical semigroups (pdf), supervised by Professors Maria Alberich Carramiñana and Alejandro Melle Hernández, on 11 July 2022 within the UCM doctoral program in Mathematical Research. Currently, he is a post-doctoral researcher at Instituto de Matemáticas Universidad de Granada (IMAG), through a postdoctoral contract in the Maria de Maeztu Programme for Centres of Excellence.
Thesis summary
The identities between the inner invariants of a mathematical object provide patterns and restrictions that allow us to better understand its inherent properties. The main purpose of this thesis has been to offer a new understanding of the identities that appear in the interplay between analytic and topological invariants of hypersurface singularities, as well as those arising between different combinatorial invariants of numerical semigroups and their associated additive structures.
The first part of the thesis deals with analytic and topological invariants of an isolated hypersurface singularity. Two prototypical examples are the Milnor number, μ, which is a topological invariant, and the Tjurina number, τ, which is an analytic invariant. Both are closely related through the study of small perturbations of the singularity. From this point of view, it is natural to try to understand up to what extent the topology, for example encoded in μ, of the germ of a singularity constrains its analytical properties, for example τ.
My main contributions in the first part of the thesis are the following: first we provide a closed formula for the minimal Tjurina number in a fixed topological class of an irreducible plane curve in terms of topological invariants of the branch [1]. Secondly, I address a question of Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of an isolated plane curve singularity [1, 3, 4] and extend it to higher dimensions. The strategy of understanding Dimca and Greuel’s question for higher dimensional singularities was the clue to provide a complete answer to it [3]. Moreover, I show the connection of the extended question with an old standing conjecture about surface singularities posed by Durfee.
To finish the first part, I study another set of invariants, which in this case are of Hodge theoretical nature, called the spectrum. For them, we establish K. Saito’s continuous limit distribution for the spectrum of Newton non-degenerate isolated hypersurface singularities [9]. Moreover, we link Saito’s distribution problem with our generalization of Dimca and Greuel’s question. As a consequence, this provides a new way of understanding the important role of Durfee’s conjecture in the context of isolated hypersurface singularities.
The second part deals with numerical semigroups and their combinatorics. First, we address Wilf’s conjecture on numerical semigroups, which asks for a lower bound of its conductor in terms of the genus and the embedding dimension of the numerical semigroup. In this direction, we present two necessary conditions for a numerical semigroup to have negative Eliahou number, which is a number whose positivity implies Wilf’s conjecture [8]. One of our contributions to Wilf’s conjecture is to propose its extension to modules over a numerical semigroup, which provides a new insight in some related problems to Wilf’s conjecture [7]. We provide a formula for the conductor of a semimodule over a numerical semigroup with two generators [5]. As a consequence we prove the generalization of Wilf’s conjecture in this particular case and reveal some interesting symmetries in the set of gaps of a numerical semigroup with two generators [6].
Finally, we also study the value set of modules over the local ring of an irreducible plane curve singularity with one Puiseux pair providing a partial generalization of a Theorem of Bresin-sky and Teissier about the value semigroup of an irreducible plane curve singularity. As a consequence, we deduce some new features about the value set of Kähler differentials of an irreducible plane curve singularity with one Puiseux pair [2]. As a sort of conclusion, this final chapter shows how the under-standing of the combinatorics of a numerical semigroups can help in order to discover new properties of some analytical and topological invariants of curve singularities.
Highlighted publication: [3]
References
[1] M. Alberich-Carramiñana, P. Almirón, G. Blanco and A. Melle-Hernández, The minimal Tjurina number of irreducible germs of plane curve singularities, Indiana Univ. Math. J. 70 No. 4 (2021), 1211–1220.
[2] M. Alberich-Carramiñana, P. Almirón, J.J. Moyano-Fernández, Curve singularities with one Puiseux pair and value sets of modules over their local rings. Under review, preprint in arXiv pdf (2021).
[3] P. Almirón, On the quotient of Milnor and Tjurina numbers for two-dimensional isolated hypersurface singularities. Math. Nachr. 295, no. 7 (2022), 1254–1263.
[4] P. Almirón, G. Blanco, A note on a question of Dimca and Greuel, C. R. Math. Acad. Sci. Paris, Ser. I 357 (2019), 205-208.
[5] P. Almirón, J.J. Moyano-Fernández, A formula for the conductor of a semi-module of a numerical semigroup with two generators, Semigroup Forum 103, no.1 (2021), 278–285.
[6] P. Almirón, J.J. Moyano-Fernández: Supersymmetric gaps of a numerical semigroup with two generators. Communications in Algebra 50, no. 10 (2022), 4252-4268.
[7] P. Almirón, J.J. Moyano-Fernández, An Extension of the Wilf conjecture to semimodules over a numerical semigroup. Preprint in arXiv pdf (2020).
[8] P. Almirón, J.J. Moyano-Fernández, Eliahou number, Wilf function and concentration of a numerical semigroup. Quaestiones Mathematicae, 46:4 (2023), 761-774.
[9] P. Almirón, M. Schulze: Limit spectral distribution for non-degenerate hyper-surface singularities. Comptes Rendus Mathématique 360 (2022), 699-710.