Planar, rational curves over ${\mathbb F}_2$ whose only singularity is a double point

TL;DR

This paper constructs large degree planar rational curves over ${ m f F}_2$ with a single double point, extending known degree limitations from characteristic 0 to characteristic 2.

Contribution

It demonstrates the existence of high-degree rational curves with a unique double point over ${ m f F}_2$, surpassing degree constraints known in characteristic zero.

Findings

Existence of large degree rational curves with a single double point over ${ m f F}_2$

Examples of supersingular double planes included

References updated in subsequent versions

Abstract

We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of supersingular double planes added. v.3: references updated.