Linear series and existence of branched covers

TL;DR

This paper investigates the existence of branched covers of the projective line in positive characteristic using linear series and degeneration techniques, providing new algebraic results related to ramification and monodromy.

Contribution

It introduces new algebraic methods leveraging linear series and degeneration tools to determine existence and non-existence of branched covers in positive characteristic.

Findings

New criteria for existence of branched covers based on ramification indices

Results linking monodromy groups to cover existence in positive characteristic

First algebraic proofs of certain non-existence results in this context

Abstract

In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on existence and non-existence of branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy groups, and the first class of results are obtained by intrinsically algebraic and positive-characteristic arguments.