Construction of covers in positive characteristic via degeneration

TL;DR

This paper constructs specific covers of the projective line in positive characteristic where all specializations are inseparable, demonstrating limitations in building all such covers from simpler cases.

Contribution

It provides explicit examples showing that not all covers of genus zero curves can be obtained inductively from covers with fewer branch points in positive characteristic.

Findings

All specializations of the constructed covers are inseparable.

Inductive construction from fewer branch points is not always possible.

Highlights limitations in the theory of covers in positive characteristic.

Abstract

In this note we construct examples of covers of the projective line in positive characteristic such that every specialization is inseparable. The result illustrates that it is not possible to construct all covers of the generic r-pointed curve of genus zero inductively from covers with a smaller number of branch points.