Topological monodromy kernels for fundamental groups of discriminant complements

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).