Unbendable rational curves of Goursat type and Cartan type

TL;DR

This paper classifies unbendable rational curves in complex manifolds, focusing on the simplest nontrivial case where such curves are of Goursat or Cartan type, revealing their geometric origins and properties.

Contribution

It characterizes unbendable rational curves of Goursat and Cartan types in low-dimensional cases, linking them to differential equations and contact geometry.

Findings

Lines on nonsingular cubic 4-folds are Goursat type.

Lines on general quartic 5-folds are Cartan type.

Projective geometry of minimal rational tangents is crucial.

Abstract

We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension $n$ with normal bundles isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus (n-1-p)}$ for some nonnegative integer $p$. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of $p=1$ and $n \leq 5$, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role.