[Paper Review] Birationality of the tangent map for minimal rational curves

Jun-Muk Hwang, Ngaiming Mok|ArXiv.org|Apr 7, 2003

Meromorphic and Entire Functions3 references4 citations

TL;DR

This paper establishes that the tangent map associating minimal rational curves on a uniruled projective manifold to their tangent directions is birational, proving that a general minimal rational curve is uniquely determined by its tangent vector. Using differential systems and the geometry of minimal rational tangents, the authors provide a Lie-theory-free proof of the rigidity of morphisms from rational homogeneous spaces of Picard number 1 to other Fano manifolds, showing such maps are either isomorphisms or projective space quotients.

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.

Motivation & Objective

• To prove that the tangent map from the universal family of minimal rational curves to the projectivized tangent bundle is birational for any uniruled projective manifold.
• To establish that a general minimal rational curve through a general point is uniquely determined by its tangent vector.
• To provide a new, Lie-theory-free proof of the rigidity result that a surjective morphism from a rational homogeneous space of Picard number 1 to a Fano manifold of Picard number 1 must be a biholomorphism or a map to projective space.
• To investigate the structure of varieties of minimal rational tangents and their role in the rigidity of generically finite morphisms to Fano manifolds.
• To prove that non-linear varieties of minimal rational tangents at general points imply strong rigidity in deformation families of Fano manifolds of Picard number 1.

Proposed method

• The authors use the theory of differential systems on subvarieties of the projectivized tangent bundle to analyze the tangent map and its fibers.
• They reduce the general case of the tangent map's birationality to the special case where the map is dominant, leveraging results from Cho-Miyaoka-Shepherd-Barron.
• The proof relies on the geometry of minimal rational curves and their tangent directions, particularly the structure of the total variety of minimal rational tangents, denoted $\mathcal{C}$.
• The authors apply deformation theory and constructible families of special cycles in the Hilbert scheme to analyze the behavior of generically finite morphisms under deformation.
• They use the existence of étale morphisms and birational maps between fibers to construct a holomorphic family of biholomorphic maps in the deformation setting.
• The argument involves analyzing the inverse image of minimal rational tangent varieties under morphisms and using flatness and birationality to deduce the existence of holomorphic families of automorphisms.

Experimental results

Research questions

• RQ1 Is the tangent map from the universal family of minimal rational curves to the projectivized tangent bundle birational for any uniruled projective manifold?
• RQ2 Can the uniqueness of minimal rational curves through a general point be established solely from their tangent vectors?
• RQ3 Under what conditions does a deformation of a generically finite morphism to a Fano manifold of Picard number 1 admit a holomorphic family of biholomorphic maps?
• RQ4 What is the role of non-linear varieties of minimal rational tangents in the rigidity of morphisms from rational homogeneous spaces?
• RQ5 Can the rigidity of morphisms from rational homogeneous spaces of Picard number 1 to other Fano manifolds be proven without using Lie theory?

Key findings

• The tangent map $\tau: \mathcal{U} \dashrightarrow \mathbb{P}T(X)$ is birational for any uniruled projective manifold and any minimal component $\mathcal{K}$, meaning minimal rational curves are uniquely determined by their tangent vectors.
• The normalization of the variety of minimal rational tangents at a general point is smooth, a consequence of the birationality of the tangent map.
• Two distinct minimal components of a uniruled projective manifold give rise to distinct total varieties of minimal rational tangents, confirming a strong uniqueness property.
• For a deformation family of Fano manifolds of Picard number 1 with non-linear varieties of minimal rational tangents, any generically finite morphism to the family lifts to a holomorphic family of biholomorphic maps.
• A surjective morphism from a rational homogeneous space $G/P$ of Picard number 1 to a Fano manifold $X$ is either an isomorphism or $X$ is isomorphic to projective space, proven without Lie theory.