Notes on Projective, Contact, and Null Curves

TL;DR

This paper reviews classical algebraic geometry of complex projective curves and applies it to classify low-degree rational null curves in the complex quadric Q^3, highlighting the Klein correspondence.

Contribution

Provides an explicit classification of low-degree rational null curves in Q^3 using classical algebraic geometry and the Klein correspondence.

Findings

Classification of low-degree rational null curves in Q^3

Recounting classical background on projective and contact curves

Application of Klein correspondence to null curves

Abstract

These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of this note consists of recounting the classical background. The main application is the explicit classification of rational null curves of low degree in Q^3. I have recently received a number of requests for these notes, so I am posting them to make them generally available.