Families of cyclic curve coverings with maximal monodromy

TL;DR

This paper investigates the algebraic monodromy of cyclic Galois coverings of curves, providing criteria for maximality, and explores implications for the Coleman-Oort conjecture and the geometry of certain Galois cover loci.

Contribution

It introduces a new criterion for the maximality of monodromy in families of cyclic covers based on G-decomposition and degeneration techniques.

Findings

No special families of Galois covers exist for genus g ≥ 8.

Criteria for maximal monodromy in cyclic Galois coverings.

Identification of totally geodesic loci for double and triple Galois covers.

Abstract

We study the algebraic monodromy of families of cyclic Galois coverings of curves. Under a condition on the $G$-decomposition of the associated variation of Hodge structures, we prove a criterion for the maximality of the monodromy. The proof combines the genus-zero case with a degeneration argument involving Prym varieties of certain admissible coverings. As a consequence of our criterion, we show that for $g\geq 8$ there exists no special family of Galois covers of the type we consider, providing new evidence towards the Coleman-Oort conjecture. Finally, we determine when the loci of double and triple Galois covers are totally geodesic.