Monodromy Groups of Hurwitz-type Problems

TL;DR

This paper solves the monodromy problem for degree-4 covers of the projective line, generalizes to Z/N coefficients, and addresses a question about Galois covers with specific deck groups.

Contribution

It provides a complete solution to the Hurwitz monodromy problem for degree-4 covers and extends the analysis to Z/N coefficients, answering a question on Galois covers.

Findings

Monodromy of degree-4 covers is determined explicitly.

Generalization to Z/N coefficients with gcd(3,N)=1.

Application to Galois covers with (Z/N)^2:S_3 deck group.

Abstract

We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of distinct points in P^1. We determine the monodromy of pi_1(P_{2g+6}) on the points of the fiber. This turns out to be the same problem as the action of pi_1(P_{2g+6}) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, gcd(3,N)=1, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P^1 with deck group (Z/N)^2:S_3.