Rational Curves on Varieties

  • Carolina Araujo
  • János Kollár
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 12)


The aim of these notes is to give an introduction to the ideas and techniques of handling rational curves on varieties. The main emphasis is on varieties with many rational curves, which are the higher dimensional analogs of rational curves and surfaces.


Line Bundle Rational Curve Irreducible Component Normal Bundle Global Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    V. Alexeev, Moduli spaces M g,n (W) for surfaces, Higher-dimensional complex varieties (Trento. 1994), de Gruyter (Berlin, 1996), pp. 1–22.Google Scholar
  2. [2]
    M. Artin and G. Winters, Degenerate fibres and stable reduction of curves, Topology, 10 (1971), 373–383.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C. H. Clemens and P. Griffiths, The Intermediate Jacobian of the cubic three-fold, Ann. of Math., 95 (1972), 281–356.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. INES, 36 (1969). 75–109.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    T. Graber. J. Harris and J. Starr. Families of Rationally Connected Varieties. J. Arner. Math. Soc, 16 (2003), 29–55.MathSciNetGoogle Scholar
  6. [6]
    A. Grothendieck. Technique de construction et théoremes d’existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki, 1960–1961, exposé 221, Soc. Math. France (Paris, 1962).Google Scholar
  7. [7]
    A. Grothendieck, Technique de descente et théoremes d’existence en géométrie algébrique V. Les schémas de Picard: théoremes dexistence; VI. Les schémas de Picard: propriétés générates, Séminaire Bourbaki, Vol. 7 exposés 232, 236, Soc. Math. France (Paris, 1962).Google Scholar
  8. [8]
    W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in: Algebraic geometry—Santa Cruz 1995 (S. Bloch et al., eds.) Proc. Sympos. Pure Math., vol. 62, Part 2, Arner. Math. Soc. (Providence, RI, 1997), pp. 45–96.CrossRefGoogle Scholar
  9. [9]
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer (1977).Google Scholar
  10. [10]
    S. Keel and S. Mori. Quotients by groupoids, Ann. of Math., 145 (1997), 193–213.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. Kollár, Projectivity of complete moduli. J. Differential Geom., 32 (1990), 235–268.MathSciNetMATHGoogle Scholar
  12. [12]
    J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32, Springer (1996).CrossRefGoogle Scholar
  13. [13]
    J. Kollár, Quotient spaces modulo algebraic groups, Ann. of Math., 145 (1997), 33–79.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. Kollár, Rationally connected varieties over local fields, Ann. of Math., 150 (1999), 357–367.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    J. Kollár, Which are the simplest Algebraic Varieties?, Bull. Amer. Math. Soc., 38 (2001), 409–433.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. Kollár, Specialization of zero cycles, preprint math. AG/0205148.Google Scholar
  17. [17]
    J. Kollár, Y. Miyaoka and S. Mori, Rationally Connected Varieties, J. Algebraic Geom., 1 (1992), 429–448.MathSciNetMATHGoogle Scholar
  18. [18]
    J. Kollár and S. Mori, Birational geometry of algebraic varieties, with the collaboration of C. II. Clemens and A. Corti. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press (1998).CrossRefMATHGoogle Scholar
  19. [19]
    M. Kontsevich and Y. I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Coram. Math. Phys., 164 (1994), 525–562.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    A. F. Lopez, Noether-Lefschetz Theory and the Picard Group of Projective Surfaces, Mem. Arner. Math. Soc., vol. 89 (1991).Google Scholar
  21. [21]
    G. Xu, Subvarieties of General Hypersurfaces in Projective Space, J. Differential Geom., 39 (1994), 139–172.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Carolina Araujo
    • 1
  • János Kollár
    • 1
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA

Personalised recommendations