arXiv subjects

Juan J. Nuño-Ballesteros

Publications and source records attributed to Juan J. Nuño-Ballesteros.

5 recordsLinked to original sources

arXiv2026

On the topology of complex map-germs and a general Lê-Greuel formula

Consider a singular holomorphic map-germ $f: (X,\underline{0}) \to (\mathbb C,0)$ where $X$ is a singular complex analytic variety in $\mathbb C^N$, and another holomorphic map-germ $g: (X,\underline{0}) \to (\mathbb C,0)$ which is "sufficiently good" relatively to $f$. We use stratified Morse theory to determine up to homeomorphism, the topology of the Milnor fiber $F_f$ out from the slice $F_{g,f}$ and the Morse data of a Morsification of the restriction of $g$ to $F_f$. This generalizes classical results for the case where $X$ is non-singular, and it provides a general formula comparing the Euler characteristics of $F_f$ and $F_{g,f}$. Restricting to the case where the singularity of $X$ at $\underline{0}$ is isolated, the formula for the difference of the Euler characteristics becomes algebraic and easily computable, generalizing in two directions the classical Lê-Greuel formula for the Milnor number of isolated complete intersection germs (ICIS): Firstly, $X$ can have an isolated singularity, and secondly $f$ can have arbitrary critical set. This unifies several known formulae in this vein: i) Lê-Greuel for ICIS of arbitrary codimension; ii) the formula relating the Milnor number of a curve with that of a function on it, and an extension of it for surfaces; iii) the formula for determinantal singularities; iv) and the one for the image Milnor number. All of these are special cases of our general formula.

math.AG↗

arXiv2024

On the analytic structure of double and triple points in the target of finite holomorphic multi-germs

We study the analytic structure of the double and triple point spaces $M_2(f)$ and $M_3(f)$ of finite multi-germs $f\colon (X,S)\to(\mathbb{C}^{n+1},0)$, based on results of Mond and Pellikaan for the mono-germ case. We show that these spaces are Cohen-Macaulay, provided that certain dimensional conditions are satisfied, and give explicit expressions for their defining ideals in terms of those of their mono-germ branches.

math.AG↗

arXiv2023

Disentangling mappings defined on ICIS

We study germs of hypersurfaces $(Y,0)\subset (\mathbb C^{n+1},0)$ that can be described as the image of $\mathscr A$-finite mappings $f:(X,S)\rightarrow (\mathbb C^{n+1},0)$ defined on an ICIS $(X,S)$ of dimension $n$. We extend the definition of the Jacobian module given by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-Sanchis when $X=\mathbb C^n$, which controls the image Milnor number $μ_I(X,f)$. We apply these results to prove the case $n=2$ of the generalised Mond conjecture, which states that $μ_I(X,f)\geq codim_{\mathscr A_e} (X,f)$, with equality if $(Y,0)$ is weighted homogeneous.

math.AG↗

arXiv2020

Equisingularity of families of functions on isolated determinantal singularities

We study the equisingularity of a family of function germs $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$, where $(X_t,0)$ are $d$-dimensional isolated determinantal singularities. We define the $(d-1)$th polar multiplicity of the fibers $X_t\cap f_t^{-1}(0)$ and we show how the constancy of the polar multiplicities is related to the constancy of the Milnor number of $f_t$ and the Whitney equisingularity of the family.

math.AG↗

arXiv2019

The Bruce-Roberts number of a function on a hypersurface with isolated singularity

Let $(X,0)$ be an isolated hypersurface singularity defined by $ϕ\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $μ_{BR}(f,X)$ is finite. We first prove that $μ_{BR}(f,X)=μ(f)+μ(ϕ,f)+μ(X,0)-τ(X,0)$, where $μ$ and $τ$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

math.AG↗