Some 4-point Hurwitz numbers in positive characteristic

TL;DR

This paper computes specific 4-point Hurwitz numbers in positive characteristic, revealing how covers degenerate to separable covers using stable reduction techniques, extending understanding from the complex case.

Contribution

It introduces methods to compute 4-point Hurwitz numbers in characteristic p and analyzes their degeneration to separable covers, expanding the theory in positive characteristic.

Findings

Computed Hurwitz numbers for degree p covers with four branch points

Demonstrated degeneration to separable covers in many cases

Extended complex case results to positive characteristic

Abstract

In this paper, we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p. Our techniques involve related computations in the case of three branch points, and allow us to conclude in many cases that for a particular choice of degeneration, all the covers we consider degenerate to separable (admissible) covers. Starting from a good understanding of the complex case, the proof is centered on the theory of stable reduction of Galois covers.