arXiv subjects

J. J. Nuño-Ballesteros

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

13 recordsLinked to original sources

arXiv2025

Counting $D_4$ singularities in the image of a wave front

We give a formula to count the number of $D_4$ singularities in a stable frontal perturbation of a corank $2$ wave front singularity $f\colon (\mathbb{C}^3,0) \to (\mathbb{C}^4,0)$ using Mond's method of stable perturbations of map germs. For a generic germ of corank $2$ wave front $f\colon (\mathbb{C}^3,S) \to (\mathbb{C}^4,0)$, the image of a stable deformation $f_t$ of $f$ exhibits $A_k$ singularities with $k \leq 4$, their transverse intersections and the aforementioned $D_4$ singularities for $0 < |t| \ll 1$. By interpreting the image of $f_t$ as the discriminant (the image of the critical point set) of a smooth map germ $H_t\colon (\mathbb{C}^5,0) \to (\mathbb{C}^4,0)$, we define an algebra whose dimension over $\mathbb{C}$ is equal to the number of $D_4$ points in the image of $f_t$.

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arXiv2024

A proof of the Mond conjecture for wave fronts

We prove the Mond conjecture for wave fronts which states that the number of parameters of a frontal versal unfolding is less than or equal to the number of spheres in the image of a stable frontal deformation with equality if the wave front is weighted homogeneous. We give two different proofs. The first one depends on the fact that wave fronts are related to discriminants of map germs and we then use the analogous result proved by Damon and Mond in this context. The second one is based on ideas by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort Sanchis and by Nuño-Ballesteros and Fernández-Hernández. The advantage of the second approach is that most results are valid for any frontal, not only wave fronts, and thus give important tools which may be useful to prove the conjecture for frontals in general.

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arXiv2024

Singularities of mappings on ICIS and applications to Whitney equisingularity

We study germs of analytic maps $f:(X,S)\rightarrow(\mathbb{C}^p,0)$, when $X$ is an ICIS of dimension $n<p$. We define an image Milnor number, generalizing Mond's definition, $μ_I(X,f)$ and give results known for the smooth case such as the conservation of this quantity by deformations. We also use this to characterise the Whitney equisingularity of families of corank one map germs $f_t\colon(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instabilities in terms of the constancy of the $μ_I^*$-sequences of $f_t$ and the projections $π\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the ICIS given by double point space of $f_t$ in $\mathbb{C}^n\times\mathbb{C}^n$. The $μ_I^*$-sequence of a map germ consist of the image Milnor number of the map germ and all its successive transverse slices.

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arXiv2024

Thom condition and Monodromy

We give the definition of the Thom condition and we show that given any germ of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a geometric local monodromy without fixed points, provided that $f\in\mathfrak m_{X,x}^2$, where $\mathfrak m_{X,x}$ is the maximal ideal of $\mathcal O_{X,x}$. This result generalizes a well-known theorem of the second named author when $X$ is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that $X$ has maximal rectified homotopical depth at $x$ and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

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arXiv2023

Deformations of corank $1$ frontals

We develop a Thom-Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather-Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb C^3$ to $\mathbb C^4$.

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arXiv2022

A weak version of Mond's conjecture

We prove that a map germ $f:(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instability is stable if and only if $μ_I(f)=0$, where $μ_I(f)$ is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that $f$ has corank one. The proof here is also valid for corank $\ge 2$, provided that $(n,n+1)$ are nice dimensions in Mather's sense (so $μ_I(f)$ is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $\mathcal{A}_e$-codimension of $f$ is $\le μ_I(f)$, with equality if $f$ is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of $f$ is a hypersurface.

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arXiv2022

Singularities of Frontal Surfaces

We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under left-right-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the cuspidal and the transverse double point curves and prove that the frontal has finite codimension if and only if both curves are reduced. Finally, we also discuss about the frontal versions of the Marar-Mond formulas and the Mond's conjecture.

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arXiv2021

The image Milnor number and excellent unfoldings

We show three basic properties on the image Milnor number $μ_I(f)$ of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond's conjecture, which says that $μ_I(f)=0$ if and only if $f$ is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $μ_I(f_t)$ constant is excellent in Gaffney's sense. By technical reasons, in the two last properties we consider only the corank 1 case.

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arXiv2016

Multiple point spaces of finite holomorphic maps

We show that there exists a unique possible definition, with certain natural properties, of the multiple point space of a holomorphic map between complex manifolds. Our construction coincides with the double point space and the k-th multiple point space for corank one map-germs, due to Mond. We also give some interesting properties of the double point space and prove that in many cases it can be computed as the zero locus of certain quotient of ideals.

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arXiv2016

A Lê-Greuel type formula for the image Milnor number

Let $f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g:(\mathbb{C}^{n-1},0)\rightarrow (\mathbb{C}^{n},0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers $μ_I(f)+μ_I(g)$ is equal to the number of critical points of the stratified Morse function $p|_{X_s}:X_s\to\mathbb{C}$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).

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arXiv2016

Equisingularity of map germs from a surface to the plane

Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_ε$ is a stabilization of $f$. We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular. As an application, we show that a family $(X_t,0)$ is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection are constant on $t$.

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arXiv2016

A Jacobian module for disentanglements and applications to Mond's conjecture

Given a germ of holomorphic map $f$ from $\mathbb C^n$ to $\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\mathbb C$ is an upper bound for the $\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particular, if $M_y(F)$ is Cohen-Macaulay, then we have Mond's conjecture for $f$. Furthermore, if $f$ is quasi-homogeneous, then Mond's conjecture for $f$ is equivalent to the fact that $M_y(F)$ is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it suffices to prove it in a suitable family of examples.

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arXiv2014

Equisingularity of families of isolated determinantal singularities

We study the topological triviality and the Whitney equisingularity of a family of isolated determinantal singularities. On one hand, we give a Lê-Ramanujam type theorem for this kind of singularities by using the vanishing Euler characteristic. On the other hand, we extend the results of Teissier and Gaffney about the Whitney equisingularity of hypersurfaces and complete intersections, respectively, in terms of the constancy of the polar multiplicities.

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