𝑅-equivalence on low degree complete intersections

Pirutka, Alena

doi:10.1090/S1056-3911-2011-00581-X

$R$-equivalence on low degree complete intersections

Author: Alena Pirutka
Journal: J. Algebraic Geom. 21 (2012), 707-719
DOI: https://doi.org/10.1090/S1056-3911-2011-00581-X
Published electronically: November 9, 2011
MathSciNet review: 2957693
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Abstract | References | Additional Information

Abstract: Let $k$ be a function field in one variable over $\mathbb C$ or the field $\mathbb C((t))$. Let $X$ be a $k$-rationally simply connected variety defined over $k$. In this paper we show that $R$-equivalence on rational points of $X$ is trivial and that the Chow group of zero-cycles of degree zero $A_0(X)$ is zero. In particular, this holds for a smooth complete intersection of $r$ hypersurfaces in $\mathbb P^n_k$ of respective degrees $d_1,\ldots ,d_r$ with $\sum \limits _{i=1}^{r}d_i^2\leq n+1$.

References

Carolina Araujo and János Kollár, Rational curves on varieties, Higher dimensional varieties and rational points (Budapest, 2001) Bolyai Soc. Math. Stud., vol. 12, Springer, Berlin, 2003, pp. 13–68. MR 2011743, DOI https://doi.org/10.1007/978-3-662-05123-8_3
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 539–545 (French). MR 1191735

J.-L. Colliot-Thélène, Variétés presque rationnelles, leurs points rationnels et leurs dégénérescences, in “Arithmetic Algebraic Geometry, Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10–15, 2007”, 1–44, Lecture Notes in Mathematics 2009, Springer-Verlag, Berlin, Heidelberg, 2011.

Jean-Louis Colliot-Thélène, Hilbert’s Theorem 90 for $K_{2}$, with application to the Chow groups of rational surfaces, Invent. Math. 71 (1983), no. 1, 1–20. MR 688259, DOI https://doi.org/10.1007/BF01393336

Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175–229 (French). MR 450280

Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces. I, J. Reine Angew. Math. 373 (1987), 37–107. MR 870307

J.-L. Colliot-Thélène and A. N. Skorobogatov, $R$-equivalence on conic bundles of degree $4$, Duke Math. J. 54 (1987), no. 2, 671–677. MR 899411, DOI https://doi.org/10.1215/S0012-7094-87-05429-9

A.J. de Jong and J. Starr, Low degree complete intersections are rationally simply connected, preprint, 2006, available at http://www.math.sunysb.edu/$\sim$ jstarr/papers/nk1006g.pdf

P. Dèbes, J.-C. Douai, and M. Emsalem, Families de Hurwitz et cohomologie non abélienne, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 1, 113–149 (French, with English and French summaries). MR 1762340

W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI https://doi.org/10.1090/pspum/062.2/1492534

Tom Graber, Joe Harris, and Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. MR 1937199, DOI https://doi.org/10.1090/S0894-0347-02-00402-2

Marvin J. Greenberg, Rational points in Henselian discrete valuation rings, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 59–64. MR 207700

János 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-Verlag, Berlin, 1996. MR 1440180

János Kollár, Rationally connected varieties over local fields, Ann. of Math. (2) 150 (1999), no. 1, 357–367. MR 1715330, DOI https://doi.org/10.2307/121107

János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625

János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, 765–779. MR 1189503

M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244

Max Lieblich, Deformation theory and rational points on rationally connected varieties, Quadratic forms, linear algebraic groups, and cohomology, Dev. Math., vol. 18, Springer, New York, 2010, pp. 83–108. MR 2648721, DOI https://doi.org/10.1007/978-1-4419-6211-9_5

David A. Madore, Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction, J. Number Theory 128 (2008), no. 4, 926–944 (French, with English and French summaries). MR 2400051, DOI https://doi.org/10.1016/j.jnt.2007.03.009

Yu. I. Manin, Kubicheskie formy: algebra, geometriya, arifmetika, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0360592

Kapil H. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139 (1994), no. 3, 641–660. MR 1283872, DOI https://doi.org/10.2307/2118574

J. Starr, Rational points of rationally simply connected varieties, preprint, 2009, available at http://www.math.sunysb.edu/$\sim$ jstarr/papers/s$\_$01$\_$09a$\_$nocomment.pdf

Claire Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés [Specialized Courses], vol. 10, Société Mathématique de France, Paris, 2002 (French). MR 1988456

Additional Information

Alena Pirutka
Affiliation: École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
MR Author ID: 934651
Email: alena.pirutka@ens.fr

Received by editor(s): December 4, 2009
Received by editor(s) in revised form: November 23, 2010
Published electronically: November 9, 2011