Skip to main content

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

  • Chapter
Book cover Global Aspects of Complex Geometry

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Carolina Araujo and János Kollár. Rational curves on varieties. In Higher dimensional varieties and rational points (Budapest, 2001), volume 12 of Bolyai Soc. Math. Stud., pages 13–68. Springer, Berlin, 2003.

    Google Scholar 

  2. Sergei J. Arakelov. Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR Ser. Mat., 35:1269–1293, 1971.

    MATH  MathSciNet  Google Scholar 

  3. Carolina Araujo. Rationally connected varieties. In Snowbird lectures in algebraic geometry, volume 388 of Contemp. Math., pages 1–16. Amer. Math. Soc., Providence, RI, 2005.

    Google Scholar 

  4. Sebastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell. The pseudo-effectuve cone of a compact Kähler manifold and varieties of negative Kodaira dimension. preprint math.AG/0405285, May 2004.

    Google Scholar 

  5. Arnaud Beauville. Riemanian holonomy and algebraic geometry. preprint math.AG/9902110, 1999.

    Google Scholar 

  6. Fedor A. Bogomolov and Michael L. McQuillan. Rational curves on foliated varieties. IHES Preprint, 2001.

    Google Scholar 

  7. Armand Borel and Raghavan Narasimhan. Uniqueness conditions for certain holomorphic mappings. Invent. Math., 2:247–255, 1967.

    Article  MATH  MathSciNet  Google Scholar 

  8. Jean-Benoît Bost. Algebraic leaves of algebraic foliations over number fields. Publ. Math. Inst. Hautes Études Sci., (93):161–221, 2001.

    MATH  MathSciNet  Google Scholar 

  9. Frédéric Campana. Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4), 25(5):539–545, 1992.

    MATH  MathSciNet  Google Scholar 

  10. Luca Chiantini and Ciro Ciliberto. Weakly defective varieties. Trans. Amer. Math. Soc., 354(1):151–178 (electronic), 2002.

    Article  MATH  MathSciNet  Google Scholar 

  11. Herbert Clemens, János Kollár, and Shigefumi Mori. Higher-dimensional complex geometry. Astérisque, (166):144 pp. (1989), 1988.

    Google Scholar 

  12. César Camacho and Alcides Lins Neto. Geometric theory of foliations. Birkhäuser Boston Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman.

    MATH  Google Scholar 

  13. Koji Cho, Yoichi Miyaoka, and Nicholas I. Shepherd-Barron. Characterizations of projective space and applications to complex symplectic manifolds. In Higher dimensional birational geometry (Kyoto, 1997), volume 35 of Adv. Stud. Pure Math., pages 1–88. Math. Soc. Japan, Tokyo, 2002.

    MATH  MathSciNet  Google Scholar 

  14. Koji Cho and Ei-ichi Sato. Smooth projective varieties with the ample vector bundle Λ2 T X in any characteristic. J. Math. Kyoto Univ., 35(1):1–33, 1995.

    MATH  MathSciNet  Google Scholar 

  15. Olivier Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.

    MATH  Google Scholar 

  16. Hubert Flenner. Restrictions of semistable bundles on projective varieties. Comment. Math. Helv., 59(4):635–650, 1984.

    MATH  MathSciNet  Google Scholar 

  17. Tom Graber, Joe Harris, and Jason Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57–67 (electronic), 2003.

    Article  MATH  MathSciNet  Google Scholar 

  18. Alexander Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math., (28):255, 1966.

    MATH  MathSciNet  Google Scholar 

  19. Rajendra V. Gurjar and De-Qi Zhang. π1 of smooth points of a log del Pezzo surface is finite. I. J. Math. Sci. Univ. Tokyo, 1(1):137–180, 1994.

    MATH  MathSciNet  Google Scholar 

  20. Robin Hartshorne. Cohomological dimension of algebraic varieties. Ann. of Math. (2), 88:403–450, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  21. Robin Hartshorne. Ample vector bundles on curves. Nagoya Math. J., 43:73–89, 1971.

    MATH  MathSciNet  Google Scholar 

  22. Joe Harris. Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. A first course, Corrected reprint of the 1992 original.

    Google Scholar 

  23. Jun-Muk Hwang and Stefan Kebekus. Geometry of chains of minimal rational curves. J. Reine Angew. Math., 584:173–194, 2005.

    MATH  MathSciNet  Google Scholar 

  24. Jun-Muk Hwang, Stefan Kebekus, and Thomas Peternell. Holomorphic maps onto varieties of non-negative Kodaira dimension. J. Alg. Geom., posted April 21, 2005, PII S 1056-3911(05)00411-X (to appear in print). Preprint math.AG/0307220, Juli 2003.

    Google Scholar 

  25. Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997.

    Google Scholar 

  26. Jun-Muk Hwang and Ngaiming Mok. Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent. Math., 131(2):393–418, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  27. Jun-Muk Hwang and Ngaiming Mok. Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent. Math., 136(1):209–231, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  28. Jun-Muk Hwang and Ngaiming Mok. Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1. J. Math. Pures Appl. (9), 80(6):563–575, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  29. Jun-Muk Hwang and Ngaiming Mok. Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles. J. Algebraic Geom., 12(4):627–651, 2003.

    MATH  MathSciNet  Google Scholar 

  30. Jun-Muk Hwang and Ngaiming Mok. Birationality of the tangent map for minimal rational curves. Asian J. Math., 8(1):51–63, 2004.

    MATH  MathSciNet  Google Scholar 

  31. Andreas Höring. Uniruled varieties with splitting tangent bundle. preprint math.AG/0505327, May 2005.

    Google Scholar 

  32. Jun-Muk Hwang. Rigidity of homogeneous contact manifolds under Fano deformation. J. Reine Angew. Math., 486:153–163, 1997.

    MATH  MathSciNet  Google Scholar 

  33. Jun-Muk Hwang. Stability of tangent bundles of low-dimensional Fano manifolds with Picard number 1. Math. Ann., 312(4):599–606, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  34. Jun-Muk Hwang. Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve. Duke Math. J., 101(1):179–187, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  35. Jun-Muk Hwang. Geometry of minimal rational curves on Fano manifolds. In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pages 335–393. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. Available on the ICTP web site at http://www.ictp.trieste.it/~pub_off/services.

    Google Scholar 

  36. Jun-Muk Hwang. Hecke curves on the moduli space of vector bundles over an algebraic curve. In Algebraic geometry in East Asia (Kyoto, 2001), pages 155–164. World Sci. Publishing, River Edge, NJ, 2002.

    Google Scholar 

  37. Shigeru Iitaka. Algebraic geometry, volume 76 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. An introduction to birational geometry of algebraic varieties, North-Holland Mathematical Library, 24.

    Google Scholar 

  38. Stefan Kebekus. Lines on contact manifolds. J. Reine Angew. Math., 539:167–177, 2001.

    MATH  MathSciNet  Google Scholar 

  39. Stefan Kebekus. Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron. In Complex geometry (Göttingen, 2000), pages 147–155. Springer, Berlin, 2002.

    Google Scholar 

  40. Stefan Kebekus. Families of singular rational curves. J. Algebraic Geom., 11(2):245–256, 2002.

    MATH  MathSciNet  Google Scholar 

  41. Stefan Kebekus. Projective bundles of singular plane cubics. Math. Nachr., 242:119–131, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  42. Stefan Kebekus. Holomorphe Abbildungen auf Mannigfaltigkeiten mit nicht-negativer Kodaira-Dimension. In Y. Tschinkel, editor, Mathematisches Institut Georg-August-Universität Göttingen Seminars 2003/2004, pages 157–166. Universitätsverlag der Georg-August-Universität Göttingen, 2004.

    Google Scholar 

  43. Stefan Kebekus. Lines on complex contact manifolds. II. Compos. Math., 141(1):227–252, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  44. Stefan Kebekus and Sándor J. Kovács. Are rational curves determined by tangent vectors? Ann. Inst. Fourier (Grenoble), 54(1):53–79, 2004.

    MATH  MathSciNet  Google Scholar 

  45. Stefan Kebekus and Sándor J. Kovács. Families of canonically polarized a a varieties over surfaces. preprint math.AG/0511378, November 2005. To appear in Invent. Math.

    Google Scholar 

  46. János Kollár and Shigefumi Mori. Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.

    MATH  Google Scholar 

  47. Seán Keel and James McKernan. Rational curves on quasi-projective surfaces. Mem. Amer. Math. Soc., 140(669):viii+153, 1999.

    MathSciNet  Google Scholar 

  48. János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rational connectedness and boundedness of Fano manifolds. J. Differential Geom., 36(3):765–779, 1992.

    MATH  MathSciNet  Google Scholar 

  49. János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rationally connected varieties. J. Algebraic Geom., 1(3):429–448, 1992.

    MATH  MathSciNet  Google Scholar 

  50. János Kollár. Extremal rays on smooth threefolds. Ann. Sci. École Norm. Sup. (4), 24(3):339–361, 1991.

    MATH  Google Scholar 

  51. János Kollár, editor. Flips and abundance for algebraic threefolds. Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992).

    MATH  Google Scholar 

  52. János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1996.

    MATH  Google Scholar 

  53. Sándor J. Kovács. Algebraic hyperbolicity of fine moduli spaces. J. Algebraic Geom., 9(1):165–174, 2000.

    MATH  MathSciNet  Google Scholar 

  54. Sándor J. Kovács. Logarithmic vanishing theorems and Arakelov-Parshin boundedness for singular varieties. Compositio Math., 131(3):291–317, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  55. Sándor J. Kovács. Families of varieties of general type: the Shafarevich conjecture and related problems. In Higher dimensional varieties and rational points (Budapest, 2001), volume 12 of Bolyai Soc. Math. Stud., pages 133–167. Springer, Berlin, 2003.

    Google Scholar 

  56. Stefan Kebekus and Thomas Peternell. A refinement of Stein factorization and deformations of surjective morphisms. preprint math.AG/0508285, August 2005.

    Google Scholar 

  57. Stefan Kebekus, Thomas Peternell, Andrew J. Sommese, and Jarosław A. Wiśniewski. Projective contact manifolds. Invent. Math., 142(1):1–15, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  58. Stefan Kebekus, Luis Solá Conde, and Matei Toma. Rationally connected a foliations after Bogomolov and McQuillan. J. Alg. Geom., posted on May 25, 2006, PII S 1056-3911(06)00435-8 (to appear in print).

    Google Scholar 

  59. Adrian Langer. Addendum to: “Semistable sheaves in positive characteristic” [Ann. of Math. (2) 159 (2004), no. 1, 251–276; mr 2051393]. Ann. of Math. (2), 160(3):1211–1213, 2004.

    MathSciNet  Google Scholar 

  60. Adrian Langer. Semistable sheaves in positive characteristic. Ann. of Math. (2), 159(1):251–276, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  61. Robert Lazarsfeld. A Barth-type theorem for branched coverings of projective space. Math. Ann., 249(2):153–162, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  62. Robert Lazarsfeld. Positivity in algebraic geometry. II, volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals.

    MATH  Google Scholar 

  63. Kenji Matsuki. Introduction to the Mori program. Universitext. Springer-Verlag, New York, 2002.

    MATH  Google Scholar 

  64. Yoichi Miyaoka. Deformations of a morphism along a foliation and applications. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 245–268. Amer. Math. Soc., Providence, RI, 1987.

    Google Scholar 

  65. Yoichi Miyaoka and Shigefumi Mori. A numerical criterion for uniruledness. Ann. of Math. (2), 124(1):65–69, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  66. Shigefumi Mori. Projective manifolds with ample tangent bundles. Ann. of Math. (2), 110(3):593–606, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  67. Shigefumi Mori. Threefolds whose canonical bundles are not numerically effective. Ann. of Math. (2), 116(1):133–176, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  68. Yoichi Miyaoka and Thomas Peternell. Geometry of higher-dimensional algebraic varieties, volume 26 of DMV Seminar. Birkhäuser Verlag, Basel, 1997.

    MATH  Google Scholar 

  69. Vikram B. Mehta and Annamalai Ramanathan. Semistable sheaves on projective varieties and their restriction to curves. Math. Ann., 258(3):213–224, 1981/82.

    Article  MathSciNet  Google Scholar 

  70. Masayoshi Miyanishi and Shuichiro Tsunoda. The structure of open algebraic surfaces. II. In Classification of algebraic and analytic manifolds (Katata, 1982), volume 39 of Progr. Math., pages 499–544. Birkhäuser Boston, Boston, MA, 1983.

    Google Scholar 

  71. Masayoshi Miyanishi and Shuichiro Tsunoda. Noncomplete algebraic surfaces with logarithmic Kodaira dimension ∔ ∞ and with nonconnected boundaries at infinity. Japan. J. Math. (N.S.), 10(2):195–242, 1984.

    MATH  MathSciNet  Google Scholar 

  72. Alan M. Nadel. The boundedness of degree of Fano varieties with Picard number one. J. Amer. Math. Soc., 4(4):681–692, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  73. Mudumbai S. Narasimhan and S. Ramanan. Geometry of Hecke cycles. I. In C. P. Ramanujam—a tribute, volume 8 of Tata Inst. Fund. Res. Studies in Math., pages 291–345. Springer, Berlin, 1978.

    Google Scholar 

  74. A. N. Parshin. Algebraic curves over function fields. I. Izv. Akad. Nauk SSSR Ser. Mat., 32:1191–1219, 1968.

    MATH  MathSciNet  Google Scholar 

  75. Thomas Peternell and Andrew J. Sommese. Ample vector bundles and branched coverings. Comm. Algebra, 28(12):5573–5599, 2000. With an appendix by Robert Lazarsfeld, Special issue in honor of Robin Hartshorne.

    MATH  MathSciNet  Google Scholar 

  76. Miles Reid. Young person’ guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 345–414. Amer. Math. Soc., Providence, RI, 1987.

    Google Scholar 

  77. C. S. Seshadri. Fibrés vectoriels sur les courbes algébriques, volume 96 of Astérisque. Société Mathématique de France, Paris, 1982. Notes written by J.-M. Drezet from a course at the école Normale Supérieure, June 1980.

    Google Scholar 

  78. Igor R. Shafarevich. Algebraic number fields. In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pages 163–176. Inst. Mittag-Leffler, Djursholm, 1963. English translation: Amer. Math. Soc. Transl. (2) 31 (1963), 25–39.

    Google Scholar 

  79. Yum-Tong Siu. Hyperbolicity in complex geometry. In The legacy of Niels Henrik Abel, pages 543–566. Springer, Berlin, 2004.

    Google Scholar 

  80. Xiaotao Sun. Minimal rational curves on moduli spaces of stable bundles. Math. Ann., 331(4):925–937, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  81. Eckart Viehweg. Positivity of direct image sheaves and applications to families of higher dimensional manifolds. In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pages 249–284. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. Available on the ICTP web site at http://www.ictp.trieste.it/~pub_off/services.

    Google Scholar 

  82. Eckart Viehweg and Kang Zuo. On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom., 10(4):781–799, 2001.

    MATH  MathSciNet  Google Scholar 

  83. Eckart Viehweg and Kang Zuo. Base spaces of non-isotrivial families of smooth minimal models. In Complex geometry (Göttingen, 2000), pages 279–328. Springer, Berlin, 2002.

    Google Scholar 

  84. Fyodor L. Zak. Tangents and secants of algebraic varieties, volume 127 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author.

    MATH  Google Scholar 

  85. De-Qi Zhang. Logarithmic del Pezzo surfaces of rank one with contractible boundaries. Osaka J. Math., 25(2):461–497, 1988.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kebekus, S., Conde, L.S. (2006). Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications. In: Catanese, F., Esnault, H., Huckleberry, A.T., Hulek, K., Peternell, T. (eds) Global Aspects of Complex Geometry. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35480-8_10

Download citation

Publish with us

Policies and ethics