Arithmetic inflection of superelliptic curves

TL;DR

This paper investigates the inflection points of linear series on superelliptic curves, providing detailed descriptions over ramification points and exploring inflectionary varieties away from ramification, based on Newton polytope analysis.

Contribution

It offers a precise characterization of inflection points on superelliptic curves and initiates a study of associated inflectionary varieties using Newton polytope methods.

Findings

Describes inflection behavior over ramification locus.

Analyzes inflectionary varieties away from ramification.

Connects inflection points to Newton polytope properties.

Abstract

In this paper, we explore the inflectionary behavior of linear series on superelliptic curves $X$ over fields of arbitrary characteristic. Here we give a precise description of the inflection of linear series over the ramification locus of the superelliptic projection; and we initiate a study of those inflectionary varieties that parameterize the inflection points of linear series on $X$ supported away from the superelliptic ramification locus that is predicated on the behavior of their Newton polytopes.