Publications mathématiques de l'IHÉS

, Volume 114, Issue 1, pp 1–85 | Cite as

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Article

Abstract

Under suitable hypotheses, we prove that a form of a projective homogeneous variety G/P defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre’s Conjecture II in Galois cohomology for function fields over an algebraically closed field.

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References

  1. [AV02]
    D. Abramovich and A. Vistoli, Compactifying the space of stable maps, J. Am. Math. Soc., 15 (2002), 27–75. (electronic). MathSciNetMATHCrossRefGoogle Scholar
  2. [BGI71]
    P. Berthelot, A. Grothendieck, and L. Illusie (eds.), Théorie des intersections et théorème de Riemann-Roch, Springer, Berlin, 1971. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre, Lecture Notes in Mathematics, Vol. 225. MATHGoogle Scholar
  3. [BLR90]
    S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer, Berlin, 1990. MATHGoogle Scholar
  4. [BM96]
    K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., 85 (1996), 1–60. MathSciNetMATHCrossRefGoogle Scholar
  5. [Bor91]
    A. Borel, Linear Algebraic Groups, Springer, Berlin, 1991. MATHCrossRefGoogle Scholar
  6. [CC98]
    A. M. Cohen and B. N. Cooperstein, Lie incidence systems from projective varieties, Proc. Am. Math. Soc., 126 (1998), 2095–2102. MathSciNetMATHCrossRefGoogle Scholar
  7. [Che95]
    C. Chevalley, Certains schémas de groupes semi-simples, in Séminaire Bourbaki, Vol. 6, pp. 219–234, Soc. Math. France, Paris, 1995. Google Scholar
  8. [Coh95]
    A. M. Cohen, Point-line spaces related to buildings, in Handbook of Incidence Geometry, pp. 647–737, North-Holland, Amsterdam, 1995. CrossRefGoogle Scholar
  9. [CTGP04]
    J.-L. Colliot-Thélène, P. Gille, and R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J., 121 (2004), 285–341. MathSciNetMATHCrossRefGoogle Scholar
  10. [Dem65]
    M. Demazure, Schémas en groupes réductifs, Bull. Soc. Math. Fr., 93 (1965), 369–413. MathSciNetMATHGoogle Scholar
  11. [Dem77]
    M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math., 39 (1977), 179–186. MathSciNetMATHCrossRefGoogle Scholar
  12. [dJS05a]
    A. J. de Jong and J. Starr, Almost proper git-stacks and discriminant avoidance. preprint, available http://www.math.columbia.edu/~dejong/, 2005.
  13. [dJS05b]
    A. J. de Jong and J. Starr, A remark on isotrivial families. preprint, available http://www.math.columbia.edu/~dejong/, 2005.
  14. [dJS06]
    A. J. de Jong and J. Starr, Low degree complete intersections are rationally simply connected. In preparation, 2006. Google Scholar
  15. [Elk73]
    R. Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup., 6 (1974), 553–603. MathSciNetGoogle Scholar
  16. [Fal03]
    G. Faltings, Finiteness of coherent cohomology for proper fppf stacks, J. Algebr. Geom., 12 (2003), 357–366. MathSciNetMATHCrossRefGoogle Scholar
  17. [FP97]
    W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic geometry—Santa Cruz 1995. Proc. Sympos. Pure Math., vol. 62, pp. 45–96, Amer. Math. Soc., Providence, 1997. Google Scholar
  18. [GHMS05]
    T. Graber, J. Harris, B. Mazur, and J. Starr, Rational connectivity and sections of families over curves, Ann. Sci. École Norm. Sup. (4), 38 (2005), 671–692. MathSciNetMATHGoogle Scholar
  19. [GHS03]
    T. Graber, J. Harris, and J. Starr, Families of rationally connected varieties, J. Am. Math. Soc., 16 (2003), 57–67 (electronic). MathSciNetMATHCrossRefGoogle Scholar
  20. [Gro62]
    A. Grothendieck, Fondements de la Géométrie Algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.]. Secrétariat mathématique, Paris, 1962. Google Scholar
  21. [Gro63]
    A. Grothendieck, Éléments de géométrie algébrique. III. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math., 17 (1963), 137–223. http://www.numdam.org/item?id=PMIHES_1961__11__5_0. MATHGoogle Scholar
  22. [Gro67]
    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math., 32 (1967), 5–361. http://www.numdam.org/item?id=PMIHES_1965__24__5_0. CrossRefGoogle Scholar
  23. [Har71]
    G. Harder, Semisimple group schemes over curves and automorphic functions, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 307–312, Gauthier-Villars, Paris, 1971. Google Scholar
  24. [Har75]
    G. Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, J. Reine Angew. Math., 274/275 (1975), 125–138. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III. MathSciNetCrossRefGoogle Scholar
  25. [Har77]
    R. Hartshorne, Algebraic geometry. Springer, New York, 1977. Graduate Texts in Mathematics, No. 52. MATHGoogle Scholar
  26. [Hir64]
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math., 79 (1964), 109–203. MathSciNetMATHCrossRefGoogle Scholar
  27. [Hir64a]
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math., 79 (1964), 205–326. MathSciNetCrossRefGoogle Scholar
  28. [HS05]
    J. Harris and J. Starr, Rational curves on hypersurfaces of low degree. II, Compos. Math., 141 (2005), 35–92. MathSciNetMATHCrossRefGoogle Scholar
  29. [HT06]
    B. Hassett and Y. Tschinkel, Weak approximation over function fields, Invent. Math., 163 (2006), 171–190. MathSciNetMATHCrossRefGoogle Scholar
  30. [Ill71]
    L. Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, vol. 239, Springer, Berlin, 1971. MATHGoogle Scholar
  31. [Kem76]
    G. R. Kempf, Linear systems on homogeneous spaces, Ann. of Math. (2), 103 (1976), 557–591. MathSciNetMATHCrossRefGoogle Scholar
  32. [Kle66]
    S. L. Kleiman, Toward a numerical criterion of ampleness, Ann. of Math. (2), 84 (1966), 293–344. MathSciNetMATHCrossRefGoogle Scholar
  33. [KM76]
    F. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., 39 (1976), 19–55. MathSciNetMATHGoogle Scholar
  34. [KM97]
    S. Keel and S. Mori, Quotients by groupoids, Ann. of Math. (2), 145 (1997), 193–213. MathSciNetMATHCrossRefGoogle Scholar
  35. [KMM92a]
    J. Kollár, Y. Miyaoka, and S. Mori, Rational connectedness and boundedness of Fano manifolds, J. Differ. Geom., 36 (1992), 765–779. MATHGoogle Scholar
  36. [KMM92b]
    J. Kollár, Y. Miyaoka, and S. Mori, Rationally connected varieties, J. Algebr. Geom., 1 (1992), 429–448. MATHGoogle Scholar
  37. [Kol96]
    J. Kollár, Rational curves on algebraic varieties, 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], vol. 32, Springer, Berlin, 1996. Google Scholar
  38. [Kon95]
    M. Kontsevich, Enumeration of rational curves via torus actions, in The Moduli Space of Curves. Progr. Math., vol. 129, pp. 335–368, Birkhäuser, Boston, 1995. CrossRefGoogle Scholar
  39. [Lan52]
    S. Lang, On quasi algebraic closure, Ann. of Math. (2), 55 (1952), 373–390. MathSciNetMATHCrossRefGoogle Scholar
  40. [Lie06]
    M. Lieblich, Remarks on the stack of coherent algebras. Int. Math. Res. Not., 12 (2006), 75273. MathSciNetGoogle Scholar
  41. [LMB00]
    G. Laumon and L. Moret-Bailly, Champs algébriques, 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], vol. 39, Springer, Berlin, 2000. MATHGoogle Scholar
  42. [Mat80]
    H. Matsumura, Commutative algebra, Mathematics Lecture Note Series, vol. 56, 2nd ed., Benjamin/Cummings, Reading, 1980. MATHGoogle Scholar
  43. [Mil80]
    J. S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, 1980. MATHGoogle Scholar
  44. [MR04]
    R. J. Marsh and K. Rietsch, Parametrizations of flag varieties, Represent. Theory, 8 (2004), 212–242. MathSciNetMATHCrossRefGoogle Scholar
  45. [Mum70]
    D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. MATHGoogle Scholar
  46. [Ols05]
    M. C. Olsson, On proper coverings of Artin stacks, Adv. Math., 198 (2005), 93–106. MathSciNetMATHCrossRefGoogle Scholar
  47. [Ols06]
    M. C. Olsson, Hom-stacks and restriction of scalars, Duke Math. J., 134 (2006), 139–164. MathSciNetMATHCrossRefGoogle Scholar
  48. [Pop74]
    V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings, Izv. Akad. Nauk SSSR, Ser. Mat., 38 (1974), 294–322. MathSciNetMATHGoogle Scholar
  49. [Slo80]
    P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980. MATHGoogle Scholar
  50. [Spr98]
    T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, 2nd ed., Birkhäuser, Boston, 1998. MATHCrossRefGoogle Scholar
  51. [Sta04]
    J. Starr, Hypersurfaces of low degree are rationally simply-connected. Preprint, 2004. Google Scholar
  52. [Sta06]
    J. Starr, Artin’s axioms, composition and moduli spaces. Preprint, 2006. Google Scholar

Copyright information

© IHES and Springer-Verlag 2011

Authors and Affiliations

  • A. J. de Jong
    • 1
  • Xuhua He
    • 2
  • Jason Michael Starr
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsThe Hong Kong University of Science and TechnologyClear Water BayHong Kong
  3. 3.Department of MathematicsStony Brook UniversityStony BrookUSA

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