Birationality of the tangent map for minimal rational curves

By Jun-Muk Hwang, N. Mok. 2003

107 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/94dc6029ecb75a540e036731ee70c097ebaf79ac

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In Applied Mathematics

1. Carolina Araujo
2. Francesco Russo
3. Jun-Muk Hwang
4. S. Ramanan
5. N. Mok

In Geometry and Topology

1. Jun-Muk Hwang
2. Carolina Araujo
3. Francesco Russo
4. N. Mok
5. S. Ramanan
6. S. Druel
7. Aron Simis

Citations in the corpus

• Varieties with quadratic entry locus, I (2007, background): “bra Comutativa e Geometria Algebrica. 2 FRANCESCO RUSSO surprising since the family of lines on such an X ⊂ PN is the minimal covering family of rational curves in the sense of Mori; see [Mo], [HM1], [HM2], [HM3], the surveys [H1], [H2] and the recent [HK] for many interesting connections between th”
• Base manifolds for fibrations of projective irreducible symplectic manifolds (2007, background): “ 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 o”
• Hecke curves and Hitchin discriminant (2003, background): “We recall the following result from [6].”
• On Fano foliations (2011, unknown): “The map τx : H̃x → Cx is in fact the normalization morphism by [32,31].”
• Recognizing Certain Rational Homogeneous Manifolds of Picard Number 1 from their Varieties of Minimal Rational Tangents (2006, background): “When X is of Picard number 1, as suggested by [8,9], ideally geometric characterizations should depend on varieties of minimal rational tangents on a nonempty open subset.”
• Rational curves of minimal degree and characterizations of projective spaces (2006, methodology): “1. ([Keb02]) τx : Hx → Cx is a finite morphism, 2. ([ HM04 ]) τx : Hx → Cx is birational, and thus 3. τx : Hx → Cx is the normalization.”
• Rigidity of rational homogeneous spaces (2006, background): “For the other aspects, we refer the reader to [11], [14] and [18].”