Zariski Density of Monodromy Groups via a Picard–Lefschetz Type Formula. (20th February 2017)

• Record Type: Journal Article
• Authors: Xu, Jinxing
• Is Part Of: International mathematics research notices. Volume 2018:Number 11(2018)
• Page Start: 3556
• Page End: 3586
• Publication Date: 2017-02-20
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw342
• Languages: English
• ISSNs: 1073-7928

Abstract

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 eigenspace of the middle cohomology of the fiber. We prove the monodromy group is Zariski dense in the corresponding linear group. As an application, we show the fundamental group of the moduli space of hyperplane arrangements is large. It can be viewed as a degenerate analogy of Carlson-Toledo's result about the monodromy groups of smooth hypersurfaces [3 ]. The main ingredient in the proof is a Picard–Lefschetz type formula for a suitable degeneration of this family.