Bifurcation braid monodromy of plane curves

TL;DR

This paper investigates the bifurcation braid monodromy of plane curves within algebraic geometry and singularity theory, comparing it to geometric monodromy and exploring implications for knot theory and mapping class groups.

Contribution

It introduces the bifurcation braid monodromy for plane curves, identifies its generators, and analyzes its relationship with geometric monodromy and knotted geometric monodromy.

Findings

Generators of the bifurcation braid monodromy are explicitly given.

The relationship between bifurcation braid monodromy and geometric monodromy is clarified.

Implications for the unfaithfulness of geometric monodromy are discussed.

Abstract

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy taking values in the Zariski resp. Artin braid group. In both cases we give the generators of the image. These results are compared with the corresponding geometric monodromy. It takes values in the mapping class group of braided surfaces. Our final result gives a precise statement about the interdependence of the two monodromy maps. Our study concludes with some implication with regard to the unfaithfulness of the geometric monodromy and the - yet unexploited - knotted geometric monodromy, which takes the ambient space into account.