Compactifications of Hurwitz spaces

TL;DR

This paper develops new modular compactifications of Hurwitz spaces by permitting branch point collisions, providing well-behaved models for low degrees and connecting to existing theories of admissible covers and hyperelliptic curves.

Contribution

It introduces novel compactifications of Hurwitz spaces through controlled branch point collisions, unifying and extending previous constructions.

Findings

Compactifications are well-behaved for degrees 2 and 3.

Recover existing spaces of twisted admissible covers.

Connects to hyperelliptic curve spaces.

Abstract

We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus $g$ curves expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These compactifications are obtained by allowing the branch points of the covers to collide to a variable extent. They are very well-behaved if $d = 2, 3$, or if relatively few collisions are allowed. We recover as special cases the spaces of twisted admissible covers of Abramovich, Corti and Vistoli and the spaces of hyperelliptic curves of Fedorchuk.