Rational points of rationally simply connected varieties over global function fields
Annales Henri Lebesgue, Volume 3 (2020), pp. 1399-1417.

Metadata

Keywords rationally simply connected varieties, rational points, degenerations

Abstract

For a complex projective manifold that is rationally connected, resp. rationally simply connected, every finite subset is connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally connected. We prove that a projective scheme over a global function field has a rational point if it deforms to a rationally simply connected variety in characteristic 0 with vanishing elementary obstruction. This gives new, uniform proofs over these fields of the Period-Index Theorem, the quasi-split case of Serre’s “Conjecture II”, and Lang’s C 2 property.


References

[BCTS08] Borovoi, Mikhail; Colliot-Thélène, Jean-Louis; Skorobogatov, Alexei N. The elementary obstruction and homogeneous spaces, Duke Math. J., Volume 141 (2008) no. 2, pp. 321-364 | DOI | MR | Zbl

[CTS87] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques La descente sur les variétés rationnelles. II, Duke Math. J., Volume 54 (1987) no. 2, pp. 375-492 | DOI | Zbl

[Del73] Deligne, N. Pierre et Katz Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Volume 340, Springer, 1973 (Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II))

[DeL15] DeLand, Matt Relatively very free curves and rational simple connectedness, J. Reine Angew. Math., Volume 699 (2015), pp. 1-33 | DOI | MR | Zbl

[Dem77] Demazure, Michel Automorphismes et déformations des variétés de Borel, Invent. Math., Volume 39 (1977) no. 2, pp. 179-186 | DOI | Zbl

[DG70] Demazure, Michel; Grothendieck, Alexander Schémas en groupes. II : Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, Volume 152, Springer, 1970 | Zbl

[dJ97] de Jong, Aise Johan Families of curves and alterations, Ann. Inst. Fourier, Volume 47 (1997) no. 2, pp. 599-621 | DOI | Numdam | MR | Zbl

[dJHS11] de Jong, Aise Johan; He, Xehua; Starr, Jason M. Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 114 (2011) no. 1, pp. 1-85 | DOI | MR | Zbl

[Esn03] Esnault, Hélène Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math., Volume 151 (2003) no. 1, pp. 187-191 | DOI | MR | Zbl

[Esn06] Esnault, Hélène Deligne’s integrality theorem in unequal characteristic and rational points over finite fields, Ann. Math., Volume 164 (2006) no. 2, pp. 715-730 (With an appendix by Pierre Deligne and Esnault) | DOI | MR | Zbl

[Esn07] Esnault, Hélène Coniveau over 𝔭-adic fields and points over finite fields, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 2, pp. 73-76 | DOI | MR | Zbl

[EX09] Esnault, Hélène; Xu, Chenyang Congruence for rational points over finite fields and coniveau over local fields, Trans. Am. Math. Soc., Volume 361 (2009) no. 5, pp. 2679-2688 | DOI | MR | Zbl

[Fin10] Findley, Robert Adam Rational curves in low degree hypersurfaces of Grassmannian varieties (2010) (Ph. D. Thesis) | MR

[Fuj02] Fujiwara, Kazuhiro A proof of the absolute purity conjecture (after Gabber), Algebraic geometry 2000, Azumino (Hotaka) (Advanced Studies in Pure Mathematics) Volume 36, Mathematical Society of Japan, 2002, pp. 153-183 | DOI | MR | Zbl

[GHMS05] Graber, Tom; Harris, Joe; Mazur, Barry; Starr, Jason Rational connectivity and sections of families over curves, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005), pp. 671-692 | DOI | Numdam | MR | Zbl

[Gil10] Gille, Philippe Serre’s conjecture II: a survey, Quadratic forms, linear algebraic groups, and cohomology (Developments in Mathematics) Volume 18, Springer, 2010, pp. 41-56 | DOI | MR | Zbl

[Gre66] Greenberg, Marvin J. Rational points in Henselian discrete valuation rings, Publ. Math., Inst. Hautes Étud. Sci. (1966) no. 31, pp. 563-568 | Numdam | MR

[Gro62] Grothendieck, Alexander Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, 1962 | Zbl

[Gro63] Grothendieck, Alexander Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II, Publ. Math., Inst. Hautes Étud. Sci., Volume 17 (1963) no. 1, pp. 5-80 | Numdam | Zbl

[Gro65] Grothendieck, Alexander Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas., Publ. Math., Inst. Hautes Étud. Sci., Volume 24 (1965) no. 1, pp. 5-223 | Numdam | Zbl

[Gro03] Grothendieck, Alexander Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques, Volume 3, Société Mathématique de France, 2003 (Séminaire de géométrie algébrique du Bois Marie 1960–61. [Geometric Algebra Seminar of Bois Marie 1960-61]. Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original)

[Gro05] Grothendieck, Alexander Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques, Volume 4, Société Mathématique de France, 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) | Zbl

[Har75] Harder, Günter 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 | Zbl

[Hir64] Hironaka, Heisuke Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. Math., Volume 79 (1964), p. 109-203, 205–326 | DOI | MR | Zbl

[HX09] Hogadi, Amit; Xu, Chenyang Degenerations of rationally connected varieties, Trans. Am. Math. Soc., Volume 361 (2009) no. 7, pp. 3931-3949 | DOI | MR | Zbl

[Jou77] Jouanolou, Jean-Pierre Cohomologie l-adique et fonctions L, Lecture Notes in Mathematics, Volume 589, Springer, 1977 (Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5)) | MR

[Jou83] Jouanolou, Jean-Pierre Théorèmes de Bertini et applications, Progress in Mathematics, Volume 42, Birkhäuser, 1983 | MR | Zbl

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

[Lie06] Lieblich, Max D. Remarks on the stack of coherent algebras, Int. Math. Res. Not., Volume 2006 (2006) no. 11, 75273 | MR | Zbl

[LMB00] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 39, Springer, 2000 | Zbl

[Pir12] Pirutka, Alena R-équivalence sur les familles de variétés rationnelles et méthode de la descente, J. Théor. Nombres Bordeaux, Volume 24 (2012) no. 2, pp. 461-473 | DOI | Numdam | MR | Zbl

[Roq05] Roquette, Peter The Brauer–Hasse–Noether theorem in historical perspective, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, Volume 15, Springer, 2005 | MR | Zbl

[SdJ10] Starr, Jason M.; de Jong, Aise Johan Almost proper GIT-stacks and discriminant avoidance, Doc. Math., Volume 15 (2010), pp. 957-972 | MR | Zbl

[Ses77] Seshadri, Conjeeveram S. Geometric reductivity over arbitrary base, Adv. Math., Volume 26 (1977) no. 3, pp. 225-274 | DOI | MR | Zbl

[Sko01] Skorobogatov, Alexei N. Torsors and rational points, Cambridge Tracts in Mathematics, Volume 144, Cambridge University Press, 2001 | MR | Zbl

[Zhu19] Zhu, Yi Homogeneous space fibrations over surfaces, J. Inst. Math. Jussieu, Volume 18 (2019) no. 2, pp. 293-327 | DOI | MR | Zbl