Families of singular rational curves

By Stefan Kebekus. 2000

124 citations recorded by the source corpus.

Let X be a projective variety which is covered by a family of rational curves of minimal degree. The classic bend-and-break argument of Mori asserts that if x and y are two general points, then there are at most finitely many curves in that family which contain both x and y. In this work we shed some light on the question as to whether two sufficiently general points actually define a unique curve. As an immediate corollary to the results of this paper, we give a characterization of projective spaces which improves on the known generalizations of Kobayashi-Ochiai's theorem.

Subfields: Computational Mathematics, Computational Mechanics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/933718b3b0355baf357b1ec7881deefdb49736d4

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In Computational Mathematics

In Computational Mechanics

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Citations in the corpus

Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation (2005, background): “Let τx : Kx PTx(X ) be the rational map sending a member of Kx to its tangent direction at x. This rational map is in fact a finite morphism ([ Ke , Theorem 3.4]) over its image Cx. We call τx the tangent morphism at x and Cx the variety of minimal rational tangents ([HM3]).”
Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron (2001, unknown): “The following theorem summarizes the results of [ Keb00 ] which form the centerpiece of our argumentation.”
Rational curves of minimal degree and characterizations of projective spaces (2006, background): “Property (3) follows from [Kol96, IV.2.9]. Property (4) is in [ Keb02 ].”
Recognizing Certain Rational Homogeneous Manifolds of Picard Number 1 from their Varieties of Minimal Rational Tangents (2006, background): “However, Kebekus [11] shows that a sufficiently general point x ∈ X, any K-rational curve passing through x is immersed at x, and he proves”
Cohomological characterizations of projective spaces and hyperquadrics (2007, methodology): “The map τx : Hx → Cx is in fact the normalization morphism by [ Keb02 ] and [HM04].”
On Fano foliations (2011, unknown): “The map τx : H̃x → Cx is in fact the normalization morphism by [32,31].”
Rigidity of rational homogeneous spaces (2006, background): “[20] showed that each member of Kx is immersed at x.”