Deformation Theory and Rational Points on Rationally Connected Varieties

  • Max Lieblich
Part of the Developments in Mathematics book series (DEVM, volume 18)


We give an account of some recent work on the existence of rational points on varieties over function fields, starting with basic material on deformation theory and the bend-and-break theorem. We emphasize the connection with the geometry of moduli spaces and include a sketch of the irreducibility of M g as a model. All details are relegated to the references.


Modulus Space Irreducible Component Deformation Theory Rational Curf Smooth Point 
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.
    F. Campana. Une version géométrique généralisée du théorème du produit de Nadel. Bull. Soc. Math. France, 119(4):479–493, 1991MATHMathSciNetGoogle Scholar
  2. 2.
    O. Debarre. Higher-dimensional algebraic geometry. Universitext. Springer, New York, 2001MATHGoogle Scholar
  3. 3.
    P. Deligne, D. Mumford. The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math., 36:75–109, 1969MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    W. Fulton, R. Pandharipande. Notes on stable maps and quantum cohomology. In Algebraic geometry – Santa Cruz 1995, vol. 62: Proceedings of the Symposium on Pure Mathematics, pp. 45–96. American Mathematical Society, Providence, RI, 1997Google Scholar
  5. 5.
    T. Graber, J. Harris, J. Starr. Families of rationally connected varieties. J. Am. Math. Soc., 16(1):57–67 (electronic), 2003Google Scholar
  6. 6.
    A. Grothendieck. Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert. In Séminaire Bourbaki, vol. 6, pages Exp. No. 221, 249–276. Socit Mathmatique de France, Paris, 1995Google Scholar
  7. 7.
    D. Huybrechts, M. Lehn. The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedrick Vieweg & Sohn, Braunschweig, 1997Google Scholar
  8. 8.
    J. Kollár. Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, vol. 32. Springer, Berlin, 1996Google Scholar
  9. 9.
    J. Kollár, Y. Miyaoka, S. Mori. Rational curves on Fano varieties. In Classification of irregular varieties (Trento, 1990), vol. 1515: Lecture Notes in Mathematics, pp. 100–105. Springer, Berlin, 1992Google Scholar
  10. 10.
    J. Kollár, S. Mori. Birational geometry of algebraic varieties, vol. 134: Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998 (translated from the 1998 Japanese original)Google Scholar
  11. 11.
    S. Mori. Projective manifolds with ample tangent bundles. Ann. Math. (2), 110(3):593–606, 1979Google Scholar
  12. 12.
    D. Mumford, J. Fogarty, F. Kirwan. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 3rd ed., vol. 34. Springer, Berlin, 1994Google Scholar
  13. 13.
    M. Schlessinger. Functors of Artin rings. Trans. Am. Math. Soc., 130:208–222, 1968MATHMathSciNetGoogle Scholar
  14. 14.
    J. Starr. Rational points of rationally simply connected varieties, 2009 (preprint)Google Scholar
  15. 15.
    J. Starr, A.J. de Jong, X. He. Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, 2009. arXiv:0809.5224Google Scholar

Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Padelford Hall, University of WashingtonSeattleUSA

Personalised recommendations