Irreducibility of Hurwitz spaces

TL;DR

This paper improves the understanding of the irreducibility of Hurwitz spaces, establishing new bounds for when these spaces are connected, especially for coverings with specific branching and monodromy conditions.

Contribution

It sharpens previous irreducibility results for Hurwitz spaces, extending them to cases with fewer branch points and specific ramification conditions, using explicit braid move calculations.

Findings

Hurwitz space H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} for positive genus curves.

For elliptic curves, irreducibility holds if n >= max{2,2d-6}.

Connected components are characterized for coverings with monodromy groups different from S_d when n is large.

Abstract

Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth, projective curve Y of positive genus, simply branched in n points, with full monodromy group S_d (math.AG/0205056). We sharpen this result and prove that H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y if n >= max{2,2d-6}. We extend the result to coverings simply branched in all but one point of the discriminant. Fixing the ramification multiplicities over the special point we prove that the corresponding Hurwitz space is irreducible if the number of simply branched points is >= 2d-2. We study also simply branched coverings with monodromy group different from S_d and when n is large enough determine the corresponding connected components of H_{d,n}(Y). Our results are based on explicit calculation of the braid moves associated with the standard generators of the n-strand braid group of Y.