Families of rationally simply connected varieties over surfaces and torsors for semisimple groups
Jong, A. J.1 ; He, Xuhua2 ; Starr, Jason Michael3
1 Department of Mathematics, Columbia University New York, NY, 10027 USA
2 Department of Mathematics, The Hong Kong University of Science and Technology Clear Water Bay, Kowloon Hong Kong
3 Department of Mathematics, Stony Brook University Stony Brook, NY, 11794 USA
Publications Mathématiques de l'IHÉS, Tome 114 (2011), pp. 1-85.

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.

DOI : 10.1007/s10240-011-0035-1
Jong, A. J. 1 ; He, Xuhua 2 ; Starr, Jason Michael 3

1 Department of Mathematics, Columbia University New York, NY, 10027 USA
2 Department of Mathematics, The Hong Kong University of Science and Technology Clear Water Bay, Kowloon Hong Kong
3 Department of Mathematics, Stony Brook University Stony Brook, NY, 11794 USA 
@article{PMIHES_2011__114__1_0,
     author = {Jong, A. J. and He, Xuhua and Starr, Jason Michael},
     title = {Families of rationally simply connected varieties over surfaces and torsors for semisimple groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--85},
     publisher = {Springer-Verlag},
     volume = {114},
     year = {2011},
     doi = {10.1007/s10240-011-0035-1},
     zbl = {1285.14053},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-011-0035-1/}
}
TY  - JOUR
AU  - Jong, A. J.
AU  - He, Xuhua
AU  - Starr, Jason Michael
TI  - Families of rationally simply connected varieties over surfaces and torsors for semisimple groups
JO  - Publications Mathématiques de l'IHÉS
PY  - 2011
SP  - 1
EP  - 85
VL  - 114
PB  - Springer-Verlag
UR  - http://archive.numdam.org/articles/10.1007/s10240-011-0035-1/
DO  - 10.1007/s10240-011-0035-1
LA  - en
ID  - PMIHES_2011__114__1_0
ER  - 
%0 Journal Article
%A Jong, A. J.
%A He, Xuhua
%A Starr, Jason Michael
%T Families of rationally simply connected varieties over surfaces and torsors for semisimple groups
%J Publications Mathématiques de l'IHÉS
%D 2011
%P 1-85
%V 114
%I Springer-Verlag
%U http://archive.numdam.org/articles/10.1007/s10240-011-0035-1/
%R 10.1007/s10240-011-0035-1
%G en
%F PMIHES_2011__114__1_0
Jong, A. J.; He, Xuhua; Starr, Jason Michael. Families of rationally simply connected varieties over surfaces and torsors for semisimple groups. Publications Mathématiques de l'IHÉS, Tome 114 (2011), pp. 1-85. doi : 10.1007/s10240-011-0035-1. http://archive.numdam.org/articles/10.1007/s10240-011-0035-1/

[AV02] Abramovich, D.; Vistoli, A. Compactifying the space of stable maps, J. Am. Math. Soc., Volume 15 (2002), pp. 27-75 (electronic) | DOI | MR | Zbl

[BGI71] Théorie des intersections et théorème de Riemann-Roch (Berthelot, P.; Grothendieck, A.; Illusie, L., eds.), 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 | Zbl

[BLR90] Bosch, S.; Lütkebohmert, W.; Raynaud, M. Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21, Springer, Berlin, 1990 | Zbl

[BM96] Behrend, K.; Manin, Yu. Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., Volume 85 (1996), pp. 1-60 | DOI | MR | Zbl

[Bor91] Borel, A. Linear Algebraic Groups, Springer, Berlin, 1991 | DOI | Zbl

[CC98] Cohen, A. M.; Cooperstein, B. N. Lie incidence systems from projective varieties, Proc. Am. Math. Soc., Volume 126 (1998), pp. 2095-2102 | DOI | MR | Zbl

[Che95] Chevalley, C. Certains schémas de groupes semi-simples, Séminaire Bourbaki (6), Soc. Math. France, Paris (1995), pp. 219-234

[Coh95] Cohen, A. M. Point-line spaces related to buildings, Handbook of Incidence Geometry, North-Holland, Amsterdam (1995), pp. 647-737 | DOI | Zbl

[CTGP04] Colliot-Thélène, J.-L.; Gille, P.; Parimala, R. Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J., Volume 121 (2004), pp. 285-341 | DOI | MR | Zbl

[Dem65] Demazure, M. Schémas en groupes réductifs, Bull. Soc. Math. Fr., Volume 93 (1965), pp. 369-413 | Numdam | MR | Zbl

[Dem77] Demazure, M. Automorphismes et déformations des variétés de Borel, Invent. Math., Volume 39 (1977), pp. 179-186 | DOI | MR | Zbl

[dJS05a] A. J. de Jong and J. Starr, Almost proper git-stacks and discriminant avoidance. preprint, available http://www.math.columbia.edu/~dejong/, 2005. | Zbl

[dJS05b] A. J. de Jong and J. Starr, A remark on isotrivial families. preprint, available http://www.math.columbia.edu/~dejong/, 2005.

[dJS06] A. J. de Jong and J. Starr, Low degree complete intersections are rationally simply connected. In preparation, 2006.

[Elk73] Elkik, R. Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup., Volume 6 (1974), pp. 553-603 | Numdam | MR | Zbl

[Fal03] Faltings, G. Finiteness of coherent cohomology for proper fppf stacks, J. Algebr. Geom., Volume 12 (2003), pp. 357-366 | DOI | MR | Zbl

[FP97] Fulton, W.; Pandharipande, R. Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math., 62), Amer. Math. Soc., Providence (1997), pp. 45-96 | Zbl

[GHMS05] Graber, T.; Harris, J.; Mazur, B.; Starr, J. Rational connectivity and sections of families over curves, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005), pp. 671-692 | Numdam | MR | Zbl

[GHS03] Graber, T.; Harris, J.; Starr, J. Families of rationally connected varieties, J. Am. Math. Soc., Volume 16 (2003), pp. 57-67 (electronic) | DOI | MR | Zbl

[Gro62] Grothendieck, A. Fondements de la Géométrie Algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 | Zbl

[Gro63] Grothendieck, A. Éléments de géométrie algébrique. III. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math., Volume 17 (1963), pp. 137-223 (http://www.numdam.org/item?id=PMIHES_1961__11__5_0) | Numdam | Zbl

[Gro67] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math., Volume 32 (1967), pp. 5-361 (http://www.numdam.org/item?id=PMIHES_1965__24__5_0) | DOI | Numdam | Zbl

[Har71] Harder, G. Semisimple group schemes over curves and automorphic functions, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris (1971), pp. 307-312 | Zbl

[Har75] Harder, G. Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, J. Reine Angew. Math., Volume 274/275 (1975), pp. 125-138 (Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III) | DOI | MR | Zbl

[Har77] Hartshorne, R. Algebraic geometry, Springer, New York, 1977 (Graduate Texts in Mathematics, No. 52) | Zbl

[Hir64] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math., Volume 79 (1964), pp. 109-203 | DOI | MR | Zbl

[Hir64a] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math., Volume 79 (1964), pp. 205-326 | DOI | MR | Zbl

[HS05] Harris, J.; Starr, J. Rational curves on hypersurfaces of low degree. II, Compos. Math., Volume 141 (2005), pp. 35-92 | DOI | MR | Zbl

[HT06] Hassett, B.; Tschinkel, Y. Weak approximation over function fields, Invent. Math., Volume 163 (2006), pp. 171-190 | DOI | MR | Zbl

[Ill71] Illusie, L. Complexe cotangent et déformations. I, Lecture Notes in Mathematics, 239, Springer, Berlin, 1971 | Zbl

[Kem76] Kempf, G. R. Linear systems on homogeneous spaces, Ann. of Math. (2), Volume 103 (1976), pp. 557-591 | DOI | MR | Zbl

[Kle66] Kleiman, S. L. Toward a numerical criterion of ampleness, Ann. of Math. (2), Volume 84 (1966), pp. 293-344 | DOI | MR | Zbl

[KM76] Knudsen, F. F.; Mumford, D. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand., Volume 39 (1976), pp. 19-55 | MR | Zbl

[KM97] Keel, S.; Mori, S. Quotients by groupoids, Ann. of Math. (2), Volume 145 (1997), pp. 193-213 | DOI | MR | Zbl

[KMM92a] Kollár, J.; Miyaoka, Y.; Mori, S. Rational connectedness and boundedness of Fano manifolds, J. Differ. Geom., Volume 36 (1992), pp. 765-779 | Zbl

[KMM92b] Kollár, J.; Miyaoka, Y.; Mori, S. Rationally connected varieties, J. Algebr. Geom., Volume 1 (1992), pp. 429-448 | Zbl

[Kol96] 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], 32, Springer, Berlin, 1996 | Zbl

[Kon95] Kontsevich, M. Enumeration of rational curves via torus actions, The Moduli Space of Curves (Progr. Math., 129), Birkhäuser, Boston (1995), pp. 335-368 | DOI | Zbl

[Lan52] Lang, S. On quasi algebraic closure, Ann. of Math. (2), Volume 55 (1952), pp. 373-390 | DOI | MR | Zbl

[Lie06] Lieblich, M. Remarks on the stack of coherent algebras, Int. Math. Res. Not., Volume 12 (2006), p. 75273 | MR | Zbl

[LMB00] Laumon, G.; Moret-Bailly, L. 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], 39, Springer, Berlin, 2000 | Zbl

[Mat80] Matsumura, H. Commutative algebra, Mathematics Lecture Note Series, 56, Benjamin/Cummings, Reading, 1980 | Zbl

[Mil80] Milne, J. S. Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, Princeton, 1980 | Zbl

[MR04] Marsh, R. J.; Rietsch, K. Parametrizations of flag varieties, Represent. Theory, Volume 8 (2004), pp. 212-242 | DOI | MR | Zbl

[Mum70] Mumford, D. Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970 | Zbl

[Ols05] Olsson, M. C. On proper coverings of Artin stacks, Adv. Math., Volume 198 (2005), pp. 93-106 | DOI | MR | Zbl

[Ols06] Olsson, M. C. Hom-stacks and restriction of scalars, Duke Math. J., Volume 134 (2006), pp. 139-164 | DOI | MR | Zbl

[Pop74] Popov, V. L. Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 38 (1974), pp. 294-322 | MR | Zbl

[Slo80] Slodowy, P. Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, 815, Springer, Berlin, 1980 | Zbl

[Spr98] Springer, T. A. Linear algebraic groups, Progress in Mathematics, 9, Birkhäuser, Boston, 1998 | DOI | Zbl

[Sta04] J. Starr, Hypersurfaces of low degree are rationally simply-connected. Preprint, 2004.

[Sta06] J. Starr, Artin’s axioms, composition and moduli spaces. Preprint, 2006.

Cité par Sources :