0000000000061407
AUTHOR
Juan J. Nuño-ballesteros
showing 9 related works from this author
Stable Images and Discriminants
2020
We show that the discriminant/image of a stable perturbation of a germ of finite \(\mathcal {A}\)-codimension is a hypersurface with the homotopy type of a wedge of spheres in middle dimension, provided the target dimension does not exceed the source dimension by more than one. The number of spheres in the wedge is called the discriminant Milnor number/image Milnor number. We prove a lemma showing how to calculate this number, and show that when the target dimension does not exceed the source dimension, the discriminant Milnor number and the \(\mathcal {A}\)-codimension obey the “Milnor–Tjurina relation” familiar in the case of isolated hypersurface singularities. This relation remains conj…
Manifolds and Smooth Mappings
2020
This is a preparatory chapter giving the necessary standard background on smooth and complex manifolds and maps.
Left-Right Equivalence and Stability
2020
We introduce the key equivalence relations on germs of maps, which play an important role throughout the book—right-equivalence and left-right equivalence (A-equivalence). These are induced by groups of diffeomorphisms, so equivalence classes have tangent spaces, and we calculate many examples, including some multi-germs. We introduce the notions of stability and finite determinacy, and prove Mather’s infinitesimal criterion for stability.
Classification of Stable Germs by Their Local Algebras
2020
We prove Mather’s theorem that stable germs are classified up to \(\mathscr {A}\)-equivalence by their local algebras. We sketch his calculation of the nice dimensions, together with his classification of stable germs in the nice dimensions, and prove that in the nice dimensions every stable germ is quasi-homogeneous with respect to suitable coordinates.
The Topology of the Milnor Fibration
2020
The fibration theorem for analytic maps near a critical point published by John Milnor in 1968 is a cornerstone in singularity theory. It has opened several research fields and given rise to a vast literature. We review in this work some of the foundational results about this subject, and give proofs of several basic “folklore theorems” which either are not in the literature, or are difficult to find. Examples of these are that if two holomorphic map-germs are isomorphic, then their Milnor fibrations are equivalent, or that the Milnor number of a complex isolated hypersurface or complete intersection singularity \((X, \underline {0})\) does not depend on the choice of functions that define …
Calculating the Homology of the Image
2020
We introduce the alternating homology of a space with a symmetric group action, and give a new construction of the image computing spectral sequence (ICSS), which computes the homology of the image of a finite map from the alternating homology of its multiple point spaces. We illustrate and motivate the ICSS with simple examples.
Multiple Points in the Target: The Case of Parameterised Hypersurfaces
2020
We focus on parameterised hypersurfaces, and explore the information one can obtain from the matrix of a presentation of the push-forward of the structure sheaf, through the use of Fitting ideals. We show that in a number of cases the spaces defined by the Fitting ideals are Cohen–Macaulay, extending the previously known range. We prove the Milnor–Tjurina relation for parameterised hypersurfaces whose dimension is no greater than two.
The Cone Structure Theorem
2020
Made available in DSpace on 2022-05-01T11:54:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-07-01 We consider the topological classification of finitely determined map germs f: (Rn, 0) → (Rp, 0) with f-1(0) = {0}. Associated with f we have a link diagram, which is well defined up to topological equivalence. We prove that f is topologically A-equivalent to the generalized cone of its link diagram. Centro de Ciências e Tecnologia Universidade Federal Do Cariri, 63048-080, Juazeiro do Norte Universidade Estadual Paulista (Unesp) Instituto de Biociências Letras e Ciências Exatas Campus de São José Do Rio Preto, 15054-000, São José do Rio Preto Departament de Matemàtiques Universitat …
Mond's conjecture for maps between curves
2017
A theorem by D. Mond shows that if f:(C,0)→C2,0 is finite and has has degree one onto its image (Y, 0), then the Ae-codimension is less than or equal to the image Milnor number μI(f), with equality if and only if (Y, 0) is weighted homogeneous. Here we generalize this result to the case of a map germ f:(X,0)→C2,0, where (X, 0) is a plane curve singularity.