Bifurcation set of multi-parameter families of complex curves

TL;DR

This paper extends the understanding of bifurcation sets for polynomial mappings from complex spaces, providing a comprehensive description for higher dimensions using Betti numbers and a vanishing phenomenon.

Contribution

It offers a complete characterization of bifurcation sets for multi-parameter complex polynomial families with dimensions m=k+1, generalizing previous results limited to lower dimensions.

Findings

Bifurcation set description in terms of Betti numbers.

Identification of a vanishing phenomenon in the fibers.

Extension of bifurcation theory to higher-dimensional cases.

Abstract

The problem of detecting the bifurcation set of polynomial mappings $\mathbb{ C}^m \to \mathbb{ C}^k$, $m\ge 2$, $m\ge k\ge 1$, has been solved in the case $m=2$, $k=1$ only. Its solution, which goes back to the 1970s, involves the non-constancy of the Euler characteristic of fibres. We provide a complete answer to the general case $m= k+1 \ge 3$ in terms of the Betti numbers of fibres and of a vanishing phenomenon discovered in the late 1990s in the real setting.