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HOMOGENEOUS SPACE FIBRATIONS OVER SURFACES

Part of: Families, fibrations

Published online by Cambridge University Press:  03 April 2017

Yi Zhu
Yi Zhu*
Affiliation:
Pure Mathematics, University of Waterloo, Waterloo, ON N2L3G1, Canada (yi.zhu@uwaterloo.ca)
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Abstract

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

Keywords

rationally simply connected varietiesprojective homogeneous spacesrational pointsfunction fields of algebraic surfaces

MSC classification

Primary: 14D06: Fibrations, degenerations

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 2 , March 2019 , pp. 293 - 327
Copyright
© Cambridge University Press 2017 

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References

Artin, M., Bertin, J. E., Demazure, M., Gabriel, P., Grothendieck, A., Raynaud, M. and Serre, J.-P., Schémas en groupes, Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques, Volume 1963/64 (Institut des Hautes Études Scientifiques, Paris, 1964).Google Scholar
Artin, M., Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189.Google Scholar
Białynicki-Birula, A., Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.Google Scholar
Borovoi, M., Colliot-Thélène, J.-L. and Skorobogatov, A. N., The elementary obstruction and homogeneous spaces, Duke Math. J. 141(2) (2008), 321–364.Google Scholar
Behrend, K., The lefschetz trace formula for the moduli stack of principal bundles. PhD thesis.Google Scholar
Berthelot, P., Grothendieck, A. and Illusie, L. (Eds.) Théorie des intersections et théorème de Riemann–Roch, Lecture Notes in Mathematics, Volume 225 (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.Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Volume 21 (Springer, Berlin, 1990).Google Scholar
Borel, A., Linear algebraic groups, second edn, Graduate Texts in Mathematics, Volume 126 (Springer, New York, 1991).Google Scholar
Brion, M., On automorphism groups of fiber bundles, Publ. Mat. Urug. 12 (2011), 39–66.Google Scholar
Brion, M., Which algebraic groups are Picard varieties? Sci. China Math. 58(3) (2015), 461–478.Google Scholar
Carrell, J. B., Torus actions and cohomology, in Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia Math. Sci., Volume 131, pp. 83–158 (Springer, Berlin, 2002).Google Scholar
Colliot-Thélène, J.-L., Gille, P. and Parimala, R., Arithmetic of linear algebraic groups over 2-dimensional geometric fields., Duke Math. J. 121(2) (2004), 285–341.Google Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., La descente sur les variétés rationnelles. II, Duke Math. J. 54(2) (1987), 375–492.Google Scholar
Du Bois, P., Complexe de de Rham filtré d’une variété singulière, Bull. Soc. Math. France 109(1) (1981), 41–81.Google Scholar
de Jong, A. J., He, X. and Starr, J. M., Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, Publ. Math. Inst. Hautes Études Sci. 114 (2011), 1–85.Google Scholar
de Jong, A. J. and Starr, J., Every rationally connected variety over the function field of a curve has a rational point, Amer. J. Math. 125(3) (2003), 567–580.Google Scholar
de Jong, A. J. and Starr, J., Low degree complete intersections are rationally simply connected. Preprint. 2006. Available at http://www.math.stonybrook.edu/∼jstarr/papers/nk1006g.pdf.Google Scholar
Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., Volume 62, pp. 45–96 (Amer. Math. Soc., Providence, RI, 1997).Google Scholar
Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc. 16(1) (2003), 57–67. (electronic).Google Scholar
Gille, P., Serre’s conjecture II: a survey, in Quadratic forms, linear algebraic groups, and cohomology, Dev. Math., Volume 18, pp. 41–56 (Springer, New York, 2010).Google Scholar
Grothendieck, A., Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962] (Secrétariat mathématique, Paris, 1962).Google Scholar
Grothendieck, A., Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Volume 224 (Springer, Berlin, 1971). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud.Google Scholar
Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), in Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 4 (Société Mathématique de France, Paris, 2005). Séminaire de Géométrie Algébrique du Bois Marie, 1962, Augmenté d’un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud], With a preface and edited by Yves Laszlo, Revised reprint of the 1968 French original.Google Scholar
Hartshorne, R., Algebraic geometry (Springer, New York, 1977). Graduate Texts in Mathematics, No. 52.Google Scholar
Harris, J. and Starr, J., Rational curves on hypersurfaces of low degree. II, Compos. Math. 141(1) (2005), 35–92.Google Scholar
Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Volume 29 (Cambridge University Press, Cambridge, 1990).Google Scholar
Kempf, G. R., Linear systems on homogeneous spaces, Ann. of Math. (2) 103(3) (1976), 557–591.Google Scholar
Kollár, J. and Kovács, S. J., Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23(3) (2010), 791–813.Google Scholar
Knudsen, F. F. and Mumford, D., The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’, Math. Scand. 39(1) (1976), 19–55.Google Scholar
Kollár, J., Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123(1) (1986), 11–42.Google Scholar
Kollár, J., 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], Volume 32 (Springer, Berlin, 1996).Google Scholar
Kollár, J., Rationally connected varieties and fundamental groups, in Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Soc. Math. Stud., Volume 12, pp. 69–92 (Springer, Berlin, 2003).Google Scholar
Kim, B. and Pandharipande, R., The connectedness of the moduli space of maps to homogeneous spaces, in Symplectic geometry and mirror symmetry (Seoul, 2000), pp. 187–201 (World Sci. Publ., River Edge, NJ, 2001).Google Scholar
Lang, S., On quasi algebraic closure, Ann. of Math. (2) 55 (1952), 373–390.Google Scholar
Nisnevich, Y. A., Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I Math. 299(1) (1984), 5–8.Google Scholar
Peyre, E., Counting points on varieties using universal torsors, in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progr. Math., Volume 226, pp. 61–81 (Birkhäuser Boston, Boston, MA, 2004).Google Scholar
Starr, J. and de Jong, J., Almost proper GIT-stacks and discriminant avoidance, Doc. Math. 15 (2010), 957–972.Google Scholar
Skorobogatov, A., Torsors and rational points, Cambridge Tracts in Mathematics, Volume 144 (Cambridge University Press, Cambridge, 2001).Google Scholar
Starr, J. M., Rational points of rationally simply connected varieties, in Variétés rationnellement connexes: aspects géométriques et arithmétiques, Panor. Synthèses, Volume 31, pp. 155–221 (Soc. Math. France, Paris, 2010).Google Scholar
Starr, J. and Xu, C., Rational points of rationally simply connected varieties over global function fields. Preprint. 2011.Google Scholar
Wittenberg, O., On Albanese torsors and the elementary obstruction, Math. Ann. 340(4) (2008), 805–838.Google Scholar