Existence of covers with fixed ramification in positive characteristic

TL;DR

This paper presents two simple methods for constructing covers with fixed ramification in positive characteristic and explores their implications for counting covers and understanding Hurwitz curves.

Contribution

It introduces two elementary constructions for covers with fixed ramification in positive characteristic and analyzes their impact on counting and structure of Hurwitz curves.

Findings

Computed the number of certain covers between projective lines with four branch points.

Gained insights into the structure of Hurwitz curves parametrizing these covers.

Provided elementary constructions applicable in positive characteristic.

Abstract

We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain information on the structure of the Hurwitz curve parametrizing these covers.