Are rational curves determined by tangent vectors?
Annales de l'Institut Fourier, Volume 54 (2004) no. 1, p. 53-79
Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of X is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.
Soit X une variété projective, revêtue par des courbes rationnelles, par exemple une variété de Fano sur le corps des nombres complexes. Dans cet article, nous donnons des conditions suffisantes pour que tout vecteur tangent en un point général de X soit tangent à au plus une courbe rationnelle de degré minimal. Comme conséquence immédiate, nous obtenons un critère d’irréductibilité de l’espace des courbes rationnelles de degré minimal
Classification:  14M99,  14J45,  14J99
Keywords: Fano manifold, rational curve of minimal degree
     author = {Kebekus, Stefan and Kov\'acs, S\'andor J.},
     title = {Are rational curves determined by tangent vectors?},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {1},
     year = {2004},
     pages = {53-79},
     doi = {10.5802/aif.2010},
     zbl = {1067.14023},
     mrnumber = {2069121},
     language = {en},
     url = {}
Kebekus, Stefan; Kovács, Sándor J. Are rational curves determined by tangent vectors?. Annales de l'Institut Fourier, Volume 54 (2004) no. 1, pp. 53-79. doi : 10.5802/aif.2010.

[BBI] L. Bădescu; M.C. Beltrametti; P. Ionescu; T. Peternell And F.O. Schreyer, Eds. Almost-lines and quasi-lines on projective manifolds, Complex Analysis and Algebraic Geometry, de Gruyter (2000), pp. 1-27 | MR 1760869 | Zbl 01440933

[CMS] K. Cho; Y. Miyaoka; N.I. Shepherd-Barron Characterizations of Projective Spaces and Applications (2000) (Preprint, October-December)

[Ei] D. Eisenbud Commutative Algebra with a View Toward Algebraic Geometry, Springer, Graduate Texts in Mathematics, Tome vol. 150 (1995) | MR 1322960 | Zbl 0819.13001

[Ha] R. Hartshorne Algebraic Geometry, Springer, Graduate Texts in Mathematics, Tome vol. 52 (1977) | MR 463157 | Zbl 0367.14001

[HM] J.-M. Hwang; N. Mok Automorphism groups of the spaces of minimal rational curves on Fano manifolds of Picard number 1 (Preprint (to appear)) | MR 2072766 | Zbl 1077.14054

[Hw] J.-M. Hwang Geometry of Minimial Rational Curves on Fano Manifolds, ICTP (Lecture Notes Series) Tome vol. VI (2001) | MR 1919462 | Zbl 1086.14506

[Ke1] S. Kebekus Rationale Kurven auf projektiven Mannigfaltigkeiten (German) (2001) (Habilitationsschrift, Feb.,

[Ke2] S. Kebekus Lines on Contact Manifolds II (2001) (e-print, LANL-Preprint, math.AG/0103208)

[Ke3] S. Kebekus Lines on contact manifolds, J. reine angew. Math, Tome 539 (2001), pp. 167-177 | MR 1863858 | Zbl 0983.53031

[Ke4] S. Kebekus Families of singular rational curves, J. Alg. Geom., Tome 11 (2002), pp. 245-256 | MR 1874114 | Zbl 1054.14035

[Ke5] S. Kebekus; I. Bauer, F. Catanese, Y. Kawamata, T. Peternell And Y.-T. Siu, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex Geometry, Collection of Papers dedicated to Hans Grauert, Springer (2002), pp. 147-156 | MR 1922103 | Zbl 1046.14028

[KMM] J. Kollár; Y. Miyaoka; S. Mori Rational Connectedness and Boundedness of Fano Manifolds, J. Diff. Geom., Tome 36 (1992), pp. 765-769 | MR 1189503 | Zbl 0759.14032

[Ko] J. Kollár Rational Curves on Algebraic Varieties, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Tome vol. 32 (1996) | MR 1440180 | Zbl 0877.14012