Zariski density of monodromy groups via Picard-Lefschetz type formula

TL;DR

This paper proves that the monodromy group of a universal family of cyclic covers of projective spaces, branched along hyperplanes, is Zariski dense, using a Picard-Lefschetz type formula for degenerations.

Contribution

It establishes Zariski density of monodromy groups in a new setting involving cyclic covers and hyperplane arrangements, extending previous results on hypersurfaces.

Findings

Monodromy group is Zariski dense in the linear group

Picard-Lefschetz formula for degenerations is key

General position hyperplane arrangements are considered

Abstract

For the universal family of cyclic covers of projective spaces branched along hyperplane arrangements in general position, we consider its monodromy group acting on an eigen linear subspace of the middle cohomology of the fiber. We prove the monodromy group is Zariski dense in the corresponding linear group. It can be viewed as a degenerate analogy of Carlson-Toledo's result about the monodromy groups of smooth hypersurfaces [Duke Math. J. 97(3) (1999), 621-648]. The main ingredient in the proof is a Picard-Lefschetz type formula for a suitable degeneration of this family.