Topological monodromy kernels for fundamental groups of discriminant complements

TL;DR

This paper investigates the topological monodromy kernels associated with families of curves on algebraic surfaces, revealing conditions under which these kernels are infinite or contain complex subgroups, with applications to toric surfaces and plane curves.

Contribution

It extends existing methods to analyze the monodromy kernels, showing they are infinite or contain free groups under certain conditions, especially for linear systems on specific surfaces.

Findings

Kernel is infinite if the monodromy image has finite index.

For plane curves, the kernel contains nonabelian free groups.

Applications include linear systems on toric surfaces and complete intersections.

Abstract

A linear system on a smooth complex algebraic surface gives rise to a family of smooth curves in the surface. Such a family has a topological monodromy representation valued in the mapping class group of a fiber. Extending arguments of Kuno, we show that if the image of this representation is of finite index, then the kernel is infinite. This applies in particular to linear systems on smooth toric surfaces and on smooth complete intersections. In the case of plane curves, we extend the techniques of Carlson-Toledo to show that the kernel is quite rich (e.g. it contains a nonabelian free group).