Special codimension one loci in Hurwitz spaces

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 $d\le 5$. 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.