Birationality of the Tangent Map for Minimal Rational Curves

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. §1. Introduction Let X be an irreducible uniruled projective variety. Let RatCurves n (X) be the normalized space of rational curves on X in the sense of [Ko]. For an irreducible component K of RatCurves n (X), let ρ : U → K and µ : U → X be the associated universal family morphisms. In other words, ρ is a P 1 -bundle over K and for α ∈ K, the corresponding rational curve in X is µ(ρ −1 (α)). An irreducible component K of RatCurves n (X) is a minimal component if µ is dominant and for a general point x ∈ X, µ −1 (x) is projective. Members of a minimal component are called minimal rational curves. For example, rational curves of minimal degree passing through a very general point of X are minimal rational curves. Here, 'very general' means that the point is chosen outside a countable union of subvarieties of dimension strictly smaller than the dimension of X. Denote by PT (X) the projectivization of the tangent bundle of the smooth part of X. Given a minimal component K, consider the rational mapfor α ∈ U such that x := µ(α) is a smooth point of X and ρ(α) corresponds to a rational curve C on X smooth at x. Let C ⊂ PT (X) be the proper image of τ . We call τ the tangent map of K and C the total variety of minimal rational tangents of K.

Base manifolds for fibrations of projective irreducible symplectic manifolds

“…If C * 1 is linearly degenerate in PT * x (X), then C 1 is a cone. Thus Proposition 2.2 is equivalent to [HwMo2,Proposition 13], which says that C 1 cannot be a cone unless it is a linear subspace. ✷ The next proposition is [HwMo1,Lemma 4.2].…”

“…An irreducible component K of the space of rational curves on X is called a minimal component if for a general point x ∈ X, the subscheme K x of K consisting of members passing through x is non-empty and complete. In this case, the subvariety C x of the projectivized tangent space PT x (X) consisting of the tangent directions at x of members of K x is called the variety of minimal rational tangents at x (see [HwMo2] for more details). For a general member C of K, the normalization ν :Ĉ → C ⊂ X is an immersion of P 1 .…”

“…On the one hand, the theory of varieties of minimal rational tangents describes a certain geometric structure arising from minimal rational curves at general points of a Fano manifold X with b 2 (X) = 1 (cf. [HwMo1], [HwMo2]). This geometric structure has differential geometric properties reflecting special features of the deformation theory of minimal rational curves.…”

Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications

“…It is based in a detailed study of the distribution P and its relation with the family H, which is beyond the purpose of this survey. We refer to [HM04] for a detailed account.…”

Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds

“…For prime Fanos, the study of covering families of lines is nothing but the classical aspect in the theory of the variety of minimal rational tangents, developed by Hwang and Mok in a remarkable series of papers, see e.g. [HM,HM2,HM3,Hw]. We also recall that lines contained in X play a key role in the proof of an important result due to Barth-Van de Ven and Hartshorne, cf.…”