A note on Hurwitz schemes of covers of a positive genus curve

TL;DR

This paper proves the irreducibility of the Hurwitz scheme parametrizing branched covers of a fixed positive genus Riemann surface with specific monodromy conditions, extending classical results to higher genus cases.

Contribution

It establishes the irreducibility of Hurwitz schemes for covers of positive genus curves, filling a gap in the existing literature.

Findings

Irreducibility of Hurwitz schemes for positive genus curves

Extension of classical genus zero results

Clarification of monodromy group conditions

Abstract

We prove the irreducibility of the space parametrizing branched covers of a fixed Riemann surface $B$ of degree $d$, with at least 2d branch points, and with monodromy group equal to $S_d$. The result is classical for $g(B)=0$. The result is well-known for $g(B) > 0$, but we could find no reference.