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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 Abstract 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)