On Legendrian curves in $\mathbb P^3$

TL;DR

This paper classifies smooth Legendrian curves in projective 3-space, showing they are either lines or twisted cubics if they are linearly normal, under a unique contact structure.

Contribution

It proves that linearly normal Legendrian curves in  ext{P}^3 are limited to lines and twisted cubics, providing a classification result in contact geometry.

Findings

Legendrian curves in  ext{P}^3 are either lines or twisted cubics.

Linearly normal Legendrian curves are classified under the given conditions.

The result relies on the uniqueness of the contact structure on  ext{P}^3.

Abstract

We show that if a smooth projective curve $C\subset\mathbb P^3$ (over an algebraically closed field of characteristic zero) is Legendrian with respect to a contact structure (it is well known that a contact structure on $\mathbb P^3$ is unique up to a linear automorphism) and $C$ is linearly normal (i.e., not an isomorphic linear projection of a smooth curve $C'\subset\mathbb P^n$, $n>3$, where $C'$ does not lie in a hyperplane) then $C$ is a twisted cubic or a line.