Families of singular rational curves
Author:
Stefan Kebekus
Journal:
J. Algebraic Geom. 11 (2002), 245-256
DOI:
https://doi.org/10.1090/S1056-3911-01-00308-3
Published electronically:
November 27, 2001
MathSciNet review:
1874114
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References |
Additional Information
Abstract: Let $X$ be a projective variety which is covered by a family of rational curves of minimal degrees. We give a bound on the dimension of the subfamily of singular rational curves. Among other applications, we will show that this yields a new characterization of the projective space in terms of rational curves.
[CMS00]CMS00 K. Cho, Y. Miyaoka and N.I. Shepherd-Barron. Characterizations of Projective Space and Applications to Complex Symplectic Manifolds. unpublished preprint, 2000.
[CS95]CS95 K. Cho and E. Sato. Smooth projective varieties with ample vector bundle $\bigwedge ^2 T_X$ in any characteristic. J. Math. Kyoto Univ., 35:1–33, 1995.
[Keb00]Keb00 S. Kebekus. Projective bundles of singular plane cubics. preprint math.AG/ 0009083, 2000, to appear in Math. Nachr.
[KO73]KO73 S. Kobayashi and T. Ochiai. Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ., 13:31–47, 1973.
[Kol96]K96 J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Springer, 1996.
[KS99]KS99 Y. Kachi and E. Sato. Polarized varieties whose points are joined by rational curves of small degrees. Illinois J. Math., 43(2):350–390, 1999.
[CMS00]CMS00 K. Cho, Y. Miyaoka and N.I. Shepherd-Barron. Characterizations of Projective Space and Applications to Complex Symplectic Manifolds. unpublished preprint, 2000.
[CS95]CS95 K. Cho and E. Sato. Smooth projective varieties with ample vector bundle $\bigwedge ^2 T_X$ in any characteristic. J. Math. Kyoto Univ., 35:1–33, 1995.
[Keb00]Keb00 S. Kebekus. Projective bundles of singular plane cubics. preprint math.AG/ 0009083, 2000, to appear in Math. Nachr.
[KO73]KO73 S. Kobayashi and T. Ochiai. Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ., 13:31–47, 1973.
[Kol96]K96 J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Springer, 1996.
[KS99]KS99 Y. Kachi and E. Sato. Polarized varieties whose points are joined by rational curves of small degrees. Illinois J. Math., 43(2):350–390, 1999.
Additional Information
Stefan Kebekus
Affiliation:
Institut für Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany
MR Author ID:
637173
Email:
stefan.kebekus@uni-bayreuth.de
Received by editor(s):
June 5, 2000
Published electronically:
November 27, 2001
Additional Notes:
The author gratefully acknowledges support by a Forschungs- stipendium of the Deutsche Forschungsgemeinschaft.
