Special codimension one loci in Hurwitz spaces

TL;DR

This paper studies special divisors in Hurwitz spaces, proving their extremality and rigidity for degrees up to 5, and explores their role in understanding the space's geometry and divisor theory.

Contribution

It introduces and analyzes two families of divisors in Hurwitz space, establishing their extremal and rigid properties in low degrees and their importance in computing the space's slope.

Findings

Divisors are extremal and rigid for degrees d ≤ 5.

Proved independence of boundary components in the compactification.

Provided general results on divisor theory of Hurwitz space.

Abstract

We investigate two families of divisors which we expect to play a distinguished role in the global geometry of Hurwitz space. In particular, we show that they are extremal and rigid in the small degree regime $d \leq 5$. We further show their significance in the problem of computing the sweeping slope of Hurwitz space in these degrees. In the process, we prove various general results about the divisor theory of Hurwitz space, including a proof of the independence of the boundary components of the admissible covers compactification. Some basic open questions and further directions are discussed at the end.