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

Nikita Kalinin

TL;DR

This work determines the number and form of Legendrian rational cubics in ${\mathbb C}P^3$ passing through three generic points and a line, proving there are exactly three such cubics for any holomorphic contact structure. It also classifies all Legendrian curves on a quadric surface by reducing to standard algebraic forms, and it verifies key computations with Macaulay2. The analysis relies on the ambient contact structure, its automorphism group $\mathrm{Sp}(4,\mathbb{C})$, and a detailed study of curves on quadrics and low-degree Legendrian curves. The results connect complex contact geometry to the geometry of rational curves and have implications for the twistor correspondence and minimal surface theory in four dimensions.

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.

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

TL;DR

This work determines the number and form of Legendrian rational cubics in passing through three generic points and a line, proving there are exactly three such cubics for any holomorphic contact structure. It also classifies all Legendrian curves on a quadric surface by reducing to standard algebraic forms, and it verifies key computations with Macaulay2. The analysis relies on the ambient contact structure, its automorphism group , and a detailed study of curves on quadrics and low-degree Legendrian curves. The results connect complex contact geometry to the geometry of rational curves and have implications for the twistor correspondence and minimal surface theory in four dimensions.

Abstract

We prove that the number of legendrian rational cubics in 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.

Paper Structure

This paper contains 7 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. The contact structure on CP^3
  3. Contact form automorphisms
  4. Curves on a hypersurface of degree two
  5. Legendrian curves of degrees one and two
  6. Legendrian cubics
  7. Proof of Theorem \ref{['th_contact']}

Key Result

Theorem 2.3

Each contact holomorphic form $\omega$ on ${\mathbb C} P^3$ is of the type where $a,b,c,p,q,r$ are constants and $pc+qb+ra\ne 0$. Furthermore, all such forms are equivalent under the $GL(4,{\mathbb C})$ action.

Theorems & Definitions (28)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3: kobayashi1959remarks
  • proof
  • Example 2.4
  • Proposition 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 18 more