The normal map for plane curves and pathologies in positive characteristic

TL;DR

This paper investigates the properties of the normal map for plane algebraic curves, demonstrating its birational nature, its ability to uniquely determine smooth curves of degree > 4 in certain characteristics, and exploring phenomena in high characteristic.

Contribution

It proves the birationality of the normal map and its uniqueness property for smooth curves of degree > 4 in characteristic zero and high positive characteristic, extending classical results.

Findings

Normal map is always birational.

For smooth curves of degree > 4, the normal map uniquely determines the curve.

In high characteristic, strange curves can share the same normal curve.

Abstract

We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use this fact to show that for smooth curves of degree higher than four the normal map uniquely determines the curve. Our proof works in characteristic zero and in positive characteristic higher than the degree of the curve. We notice also that in high characteristic strange curves provide examples of different plane curves with same curve of normal lines. We will reinterpret our results also in the modern terminology of bottlenecks of algebraic curves.