Legendrian curves in $\mathbb C P^3$: cubics and curves on a quadric surface

TL;DR

This paper determines the count of Legendrian rational cubics passing through specific points and a line in complex projective 3-space, and classifies Legendrian curves on a quadric surface, supported by computational verification.

Contribution

It provides the first explicit enumeration of Legendrian rational cubics under given conditions and classifies Legendrian curves on a quadric surface, advancing understanding in contact geometry.

Findings

Number of Legendrian rational cubics through three points and a line is three.

Complete classification of Legendrian curves on a quadric surface.

Computational verification using Macaulay2.

Abstract

We prove that the number of legendrian rational cubics in $\mathbb C P^3$ through three generic points and a line is three; also we classify all legendrian curves on a quadric surface. Several computations are additionally verified using Macaulay2 computer algebra system.