Roberto Giménez Conejero

Roberto Giménez Conejero
University of Valencia | UV · Department of Geometry and Topology

About

10
Publications
754
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5
Citations
Citations since 2017
10 Research Items
5 Citations
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Introduction
Singularities of mappings fron C^n to C^{n+1}. Specially: Properties of image Milnor number, properties of one-parameter unfoldings (Whitney equisingularity, exellency, topological triviality, etc.).

Publications

Publications (10)
Article
Full-text available
We give the definition of the Thom condition and we show that given any germ of complex analytic function $$f:(X,x)\rightarrow ({\mathbb {C}},0)$$ f : ( X , x ) → ( C , 0 ) on a complex analytic space X , there exists a geometric local monodromy without fixed points, provided that $$f\in {\mathfrak {m}}_{X,x}^2$$ f ∈ m X , x 2 , where $${\mathfrak...
Article
We study germs of analytic maps f:(X,S)→(Cp,0), when X is an icis of dimension n<p. We define an image Milnor number, generalizing Mond's definition, μI(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...
Preprint
Full-text available
We give a simple way to study the isotypes of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of \textsc{icis}. We study the homology of images of mappings $f_t$ that arise as deformations of complex map germs $f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)$, with $n<p$, and the behaviour of singularities (in...
Conference Paper
Full-text available
El Seminari Predoc es una comunidad de práctica organizada por los estudiantes de doctorado vinculados a la Facultat de Ciències Matemàtiques de la Universitat de València que permite dar a conocer su trabajo a toda la comunidad educativa. Este seminario tiene el objetivo de ayudar a los participantes a perfilar y mejorar sus ponencias de carácter...
Preprint
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 $\mu_I(f)=0$, where $\mu_I(f)$ 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...
Thesis
Full-text available
We have been able to prove a characterization of the Whitney equisingularity for families of corank one germs from C n to C n+1 using a few invariants. To do so, we have proven Houston's conjecture on excellent unfoldings, proving also some fundamental results of the image Milnor number. We have also expanded the theory of map germs on icis, we hav...
Preprint
Full-text available
We 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 $\mu_I^*$-sequences of $f_t$ and the projections $\pi\colon D^2(f_t)\to\mathbb{C}^n$, where $D^2(f_t)$ is the double point space in $\mathbb{C}^n\times\mathbb...
Preprint
Full-text available
We show that given any germ of complex analytic function $f\colon(X,x)\to(\mathbb{C},0)$ 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 se...
Article
We show three basic properties of the image Milnor number µI(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...
Preprint
Full-text available
We show three basic properties on the image Milnor number $\mu_I(f)$ 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 Mo...

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Projects

Projects (2)
Project
I am starting some projects to get more information, or some kind of restrictions, through the image Milnor number/A_e-codimension of a map-germ from Cn to Cn+1.
Project
The goal is to find a workable family of invariants of germs from Cn to Cn+1 that, in case of being constant, guarantee that a 1-parameter family of such germs are Whitney equisingular. We are working on some details, this is soon to be public.