On a plane section of an integral curve in positive characteristic

TL;DR

This paper proves that for integral curves in projective 3-space over an algebraically closed field of positive characteristic, the general minimal containing curve of a plane section remains irreducible, extending known characteristic zero results.

Contribution

It establishes the irreducibility of the general minimal curve containing a plane section of an integral curve in positive characteristic, generalizing prior characteristic zero results.

Findings

The general minimal curve containing the plane section is irreducible in positive characteristic.

Plane sections may not be in uniform position in positive characteristic.

The result extends classical theorems from characteristic zero to positive characteristic.

Abstract

If $C \subset P^3_k$ is an integral curve and $k$ an algebraically closed field of characteristic 0, it is known that the points of the general plane section $C \cap H$ of $C$ are in uniform position. From this it follows easily that the general minimal curve containing $C\cap H$ is irreducible. If $char k = p > 0$, the points of $C\cap H$ may not be in uniform position. However, we prove that the general minimal curve containing $C\cap H$ is still irreducible.