On Legendrian curves in ${\mathbb{P}}^3$

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 $C^{\prime }\subset {\mathbb{P}}^n$, $n>3$, where $C^{\prime }$ does not lie in a hyperplane) then $C$ is a twisted cubic or a line.

Jarosław Buczyński pointed out that the proof of the main Theorem 1.1 contains a fatal error and other results are not new. Many thanls to Jarosław and sorry for submitting a text with errors