On a singular variety associated to a polynomial mapping from $\C^n$ to $\C^{n-1}$

TL;DR

This paper constructs a singular variety associated with polynomial mappings from complex n-space to (n-1)-space and investigates its homological properties, revealing non-trivial homology in certain cases when the map is a local submersion but not a fibration.

Contribution

It introduces a new singular variety linked to polynomial maps and proves non-trivial homology properties under specific conditions, extending understanding of such mappings.

Findings

Homology of ${\ m\mathcal{V}}_G$ is non-trivial in certain cases

Non-fibration local submersions yield non-trivial intersection homology

Results extend to higher dimensions with additional conditions

Abstract

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G : \C^{n} \to \C^{n - 1}$ where $n \geq 2$. We prove that in the case $G : \C^{3} \to \C^{2}$, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in dimension two of the variety ${\mathcal{V}}_G$ is not trivial. In the case of a local submersion $G : \C^{n} \to \C^{n - 1}$ where $n \geq 4$, the result is still true with an additional condition.