Searcharxiv⌕ Search

arXiv subjects

Patricio Almirón

Publications and source records attributed to Patricio Almirón.

16 recordsLinked to original sources
arXiv2025

Topological Representations of Free Numerical Semigroups via Iterated Torus Knots

In this paper we will associate a family $\{K_1,\dots,K_l\}\subset \mathbb{S}^3$ of iterated torus knots to a given free numerical semigroup. We will describe the fundamental group of the knot complement of each knot of the family. Finally, we will show that all knots in the family have same Alexander polynomial and it coincides (up to a factor) with the Poincaré series of the free numerical semigroup. As a consequence, we will provide families of iterated torus knots with the same Alexander polynomial of an irreducible plane curve singularity but which are non-isotopic to its associated knot.

math.GT↗
arXiv2025

On the Rich Landscape of Complete Intersection Monomial Curves

The aim of this survey is to explore complete intersection monomial curves from a contemporary perspective. The main goal is to help readers understand the intricate connections within the field and its potential applications. The properties of any monomial curve singularity will be first reviewed, highlighting the interaction between combinatorial and algebraic properties. Next, we will revisit the two main characterizations of complete intersection monomial curves. One is based on deep algebraic properties given by Herzog and Kunz in 1971, while the other is based on a combinatorial approach given by Delorme in 1976. Our aim is to bridge the gap between these perspectives present in the current literature. Then, we will focus on recent advances that show an intriguing connection between numerical semigroups and Alexander polynomials of knots. Finally, we will revisit the deformation theory of these curves.

math.AC↗
arXiv2024

The miniversal deformation of certain complete intersection monomial curves

The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup -- these curves make up a notable family of complete intersection monomial curves. First, we dispense a general decomposition result of a basis of the miniversal deformation of any complete intersection monomial curve. As a consequence, we explicitly calculate this basis in the particular case of a monomial curve defined from a free semigroup. This explicit computation yields some estimates for the dimension of the moduli space of these family of curves.

math.AG↗
arXiv2023

The intrinsic topological nature of the Poincaré series of a plane curve singularity

In this paper we provide some factorization theorems of the Poincaré series $P_C$ of a plane curve singularity $C$ depending on some key values of the semigroup of values of \(C\). These results yield an iterative computation of $P_C$ in purely algebraic terms from the dual resolution graph of $C$. On the other hand, Campillo, Delgado and Gusein-Zade showed in 2003 the equality between $P_C$ and the Alexander polynomial $Δ_L$ of the corresponding link $L$. Our procedure supplies a new proof of this coincidence. More concretely, we show that our algebraic construction can be translated to the iterated toric structure of the link $L$. Additionally we show that the semigroup algebra can be defined from the fundamental group of the link exterior in the irreducible case. This gives in particular a conceptual reason for the coincidence of $P_C$ and $Δ_L$.

math.AG↗
arXiv2022

Curve singularities with one Puiseux pair and value sets of modules over their local rings

In this paper we characterize the value set $Δ$ of the $R$-modules of the form $R+zR$ for the local ring $R$ associated to a germ $ξ$ of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the module of Kähler differentials attached to $ξ$, we recover some results of Delorme. From our characterization of $Δ$ we introduce a proper subset of semimodules over the value semigroup of the ring $R$. Moreover, we provide a geometric algorithm to construct all possible semimodules in this subset for a given value semigroup.

math.AC↗
arXiv2022

The Tjurina number for Sebastiani-Thom type isolated hypersurface singularities

In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and Tjurina numbers in terms of the singularity index of any isolated hypersurface singularity, not necessarily a join of singularities.

math.AG↗
arXiv2022

Limit spectral distribution for non-degenerate hypersurface singularities

We establish Kyoji Saito's continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito's notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.

math.AG↗
arXiv2021

On the quotient of Milnor and Tjurina numbers for two-dimensional isolated hypersurface singularities

In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

math.AG↗
arXiv2021

An extension of the Wilf conjecture to semimodules over a numerical semigroup

The aim of this paper is to propose an extension of the Wilf conjecture to semimodules over a numerical semigroup through a new approach toward the solution of the Wilf conjecture on numerical semigroups. The key point is the introduction of a new invariant that we call Wilf function of a semigroup that can be also defined for a semimodule over the semigroup. We also study the relation of the Wilf function of a semigroup and the Wilf function of certain semimodules associated to a gap of the semigroup. As a consequence, we obtain an alternative conjecture which also implies the Wilf conjecture. Finally, we show that the natural generalization of Wilf conjecture to semimodules over the semigroup does not work unless we used the Wilf function of the semimodule. In this direction, we propose several questions regarding the possible extensions of Wilf-type inequalities for semimodules.

math.CO↗
arXiv2021

Supersymmetric gaps of a numerical semigroup with two generators

In this paper we introduce the new concepts of supersymmetric and self-symmetric gaps of a numerical semigroup with two generators. Those concepts are based on certain symmetries of the gaps of the semigroup with respect to their Wilf number. We prove that the set of supersymmetric and self-symmetric gaps completely determines the semigroup and we compare this set with the fundamental gaps of the semigroup.

math.CO↗
arXiv2021

Eliahou number, Wilf function and concentration of a numerical semigroup

We give an estimate of the minimal positive value of the Wilf function of a numerical semigroup in terms of its concentration. We describe necessary conditions for a numerical semigroup to have negative Eliahou number in terms of its multiplicity, concentration and Wilf function. Also, we show new examples of numerical semigroups with negative Eliahou number. In addition, we introduce the notion of highly dense numerical semigroup; this yields a new family of numerical semigroups satisfying the Wilf conjecture. Moreover, we use the Wilf function of a numerical semigroup to prove that the Eliahou number of a highly dense numerical semigroup is positive under certain additional hypothesis. In particular, these results provide new evidences in favour of the Wilf conjecture.

math.AC↗
arXiv2021

A formula for the conductor of a semimodule of a numerical semigroup with two generators

We provide an expression for the conductor $c(Δ)$ of a semimodule $Δ$ of a numerical semigroup $Γ$ with two generators in terms of the syzygy module of $Δ$ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of $Γ$, as well as a formula for $c(Δ)$ in terms of the dual semimodule of $Δ$.

math.AC↗
arXiv2020

The minimal Tjurina number of irreducible germs of plane curve singularities

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.

math.AG↗
arXiv2019

A note on a question of Dimca and Greuel

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.

math.AG↗