Birationality of the tangent map for minimal rational curves

TL;DR

This paper proves that on uniruled projective manifolds, minimal rational curves are uniquely determined by their tangent vectors, and provides a new proof that certain holomorphic maps are biholomorphic, avoiding Lie theory.

Contribution

It establishes the birationality of the tangent map for minimal rational curves and offers a novel proof of a biholomorphicity result without Lie theory.

Findings

Minimal rational curves are uniquely determined by tangent vectors.

A holomorphic map from a rational homogeneous space of Picard number 1 to a different manifold is biholomorphic.

The tangent map for minimal rational curves is birational.

Abstract

For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a holomorphic map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biholomorphic map.