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Vanishing homology and multiple-point spaces of singular maps

Grant number: 16/23906-3
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): May 01, 2017
Effective date (End): January 31, 2018
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:João Nivaldo Tomazella
Grantee:Guillermo Penafort Sanchis
Home Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
Associated research grant:14/00304-2 - Singularities of differentiable mappings: theory and applications, AP.TEM


This project is concerned with vanishing homology of the image $Z_\epsilon= f_\epsilon(\mathbb C^n)$ of perturbations $f_\epsilon$ of finite map-germs $f\colon (\mathbb C^n,0)\to(\mathbb C^p,0)$. Results in this field run parallel to the ones for Milnor fibrations of germs of complete intersections with isolated singularity (ICIS), but the highly singular nature of $Z_\epsilon$ imposes obstacles that have limited progress. The introduction of \emph{multiple-point spaces} of maps is a means to circumvent these obstacles, because they are significantly less singular and their \emph{alternating homology modules} encode the homology of $Z_\epsilon$. However, our ability to describe the multiple-point spaces depends strongly on the corank of $f$ ($\corank f_x$ is the rank of the kernel of the differential $df_x$), and developing a good scheme-theoretical description of multiple-points in the presence of points of corank $\geq 2$ is one of the hardest problems in the field. Pe\~nafort introduced modified multiple-point spaces that overcome these problems in a wide range of dimensions. The main problems to be dealt with are (A) Mond's conjecture, which is the inequality for maps-germs $\mathbb C^n\to \mathbb C^{n+1}$ corresponding to that of Milnor and Tjurina numbers for ICIS. It is only known to be true for $n=1,2$. Bobadilla, Nu\~no and Pe\~nafort have shown that it can be reduced to a discrete family of examples of increasing order. There is still no such family in the literature, and Or\'efice-Okamoto and Pe\~nafort are currently working on the search for such a family for $n=3$. (B) Connectivity results: determination --in terms of the dimensions of source, target and singular locus-- of a range of degrees where the homology of $Z_\epsilon$ is trivial, in the same vein as Kato-Matsumoto's theorem for complete intersections. This is closely related to that the study of Tjurina modifications of determinantal varieties studied by Nu\~no, Tomazella and Or\'efice-Okamoto; and (C) Extended L\^e's conjecture: Is there any injective map-germ $f\colon (\mathbb C^n,0)\to(\mathbb C^p,0)$ satisfying $\corank f>p-n$? (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
PENAFORT SANCHIS, GUILLERMO. Reflection maps. MATHEMATISCHE ANNALEN, JUL 2020. Web of Science Citations: 0.

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