Duke Mathematical Journal

Rationally connected varieties over finite fields

János Kollár and Endre Szabó

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Abstract

Let $X$ be a geometrically rational (or, more generally, separably rationally connected) variety over a finite field $K$. We prove that if $K$ is large enough, then $X$ contains many rational curves defined over $K$. As a consequence we prove that $R$-equivalence is trivial on $X$ if $K$ is large enough. These results imply the following conjecture of J.-L. Colliot-Thélène: Let $Y$ be a rationally connected variety over a number field $F$. For a prime $P$, let $Y_P$ denote the corresponding variety over the local field $F_P$. Then, for almost all primes $P$, the Chow group of 0-cycles on $Y_P$ is trivial and $R$-equivalence is trivial on $Y_P$.

Article information

Source
Duke Math. J., Volume 120, Number 2 (2003), 251-267.

Dates
First available in Project Euclid: 16 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1082138584

Digital Object Identifier
doi:10.1215/S0012-7094-03-12022-0

Mathematical Reviews number (MathSciNet)
MR2019976

Zentralblatt MATH identifier
1077.14068

Subjects
Primary: 14G15: Finite ground fields 14J20: Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] 14M20: Rational and unirational varieties [See also 14E08]
Secondary: 14C15: (Equivariant) Chow groups and rings; motives 14G20: Local ground fields

Citation

Kollár, János; Szabó, Endre. Rationally connected varieties over finite fields. Duke Math. J. 120 (2003), no. 2, 251--267. doi:10.1215/S0012-7094-03-12022-0. https://projecteuclid.org/euclid.dmj/1082138584


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References

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  • \lccJ.-L. Colliot-Thélène and J.-J. Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), 175–229.
  • \lccJ.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces, I, J. Reine Angew. Math. 373 (1987), 37–107.
  • ––––, Intersections of two quadrics and Châtelet surfaces, II, J. Reine Angew. Math. 374 (1987), 72–168.
  • \lccO. Debarre, Variétés rationnellement connexes, to appear in Séminaire Bourbaki 2001/02, exp. 905, \wwwwww-irma.u-strasbg.fr/~debarre
  • \lccH. Esnault, Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math. 151 (2003), 187–191. \CMP1 943 746
  • \lccW. Fulton and R. Pandharipande, "Notes on stable maps and quantum cohomology" in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997, 45–96.
  • \lccM. Fried and M. Jarden, Field Arithmetic, Ergeb. Math. Grenzgeb. (3) 11, Springer, Berlin, 1986.
  • \lccP. Gille, La R-équivalence sur les groupes algébriques réductifs définis sur un corps global, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 199–235.
  • ––––, $R$-équivalence sur les $G$-revêtements sur les corps locaux non archimédiens, J. Number Theory 91 (2001), 284–292.
  • \lccJ. Harris, B. Mazur, and R. Pandharipande, Hypersurfaces of low degree, Duke Math. J. 95 (1998), 125–160.
  • \lccW. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge Univ. Press, Cambridge, 1947.
  • \lccK. Kato and S. Saito, Unramified class field theory of arithmetical surfaces, Ann. of Math. (2) 118 (1983), 241–275.
  • \lccB. Kim and R. Pandharipande, "The connectedness of the moduli space of maps to homogeneous spaces" in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 187–201.
  • \lccJ. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • ––––, Rationally connected varieties over local fields, Ann. of Math. (2) 150 (1999), 357–367.
  • ––––, Fundamental groups of rationally connected varieties, Michigan Math. J. 48 (2000), 359–368.
  • ––––, Which are the simplest algebraic varieties?, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 409–433.
  • ––––, "Rationally connected varieties and fundamental groups" in Higher Dimensional Varieties and Rational Points (Budapest, 2001), ed. K. Böröczky Jr., J. Kollár, and T. Szamuely, Bolyai Soc. Math. Stud. 12, Springer, New York, 2003, 69–92.
  • ––––, letter to J. Starr, 2001.
  • \lccJ. Kollár, Y. Miyaoka, and S. Mori, Rationally connected varieties, J. Algebraic Geometry 1 (1992), 429–448.
  • \lccS. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827.
  • \lccD. Madore, Équivalence rationnelles sur les hypersurfaces cubiques sur les corps $p$-adiques, Manuscripta Math. 110 (2003), 171–185. \CMP1 962 532
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  • \lccL. Moret-Bailly, R-équivalence simultanée de torseurs: un complément à l'article de P. Gille, J. Number Theory 91 (2001), 293–296.
  • ––––, Sur la R-équivalence de torseurs sous un groupe fini, J. Number Theory 99 (2003), 383–404. \CMP1 968 460
  • \lccS. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), 593–606.
  • \lccD. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204.
  • \lccJ.-P. Serre, Corps locaux, Publ. Inst. Math. Univ. Nancago 8, Actualités Sci. Indust. 1296, Hermann, Paris, 1962.
  • \lccT. Shioda and T. Katsura, On Fermat varieties, Tôhoku Math. J. 31 (1979), 97–115.
  • \lccH. P. F. Swinnerton-Dyer, "Universal equivalence for cubic surfaces over finite and local fields" in Symposia Mathematica, Vol. 24 (Rome, 1979), Academic Press, London, 1981, 111–143. K. Böröczky Jr., J. Kollár, and T. Szamuely, Bolyai Soc. Math. Stud. 12, Springer, New York, 2003, 13–68.
  • \lccJ.-L. Colliot-Thélène, Hilbert's Theorem 90 for $K_2$, with application to the Chow groups of rational surfaces, Invent. Math 71 (1983), 1–20.
  • \lccJ.-L. Colliot-Thélène and J.-J. Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), 175–229.
  • \lccJ.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces, I, J. Reine Angew. Math. 373 (1987), 37–107.
  • ––––, Intersections of two quadrics and Châtelet surfaces, II, J. Reine Angew. Math. 374 (1987), 72–168.
  • \lccO. Debarre, Variétés rationnellement connexes, to appear in Séminaire Bourbaki 2001/02, exp. 905, \wwwwww-irma.u-strasbg.fr/~debarre
  • \lccH. Esnault, Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math. 151 (2003), 187–191. \CMP1 943 746
  • \lccW. Fulton and R. Pandharipande, "Notes on stable maps and quantum cohomology" in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997, 45–96.
  • \lccM. Fried and M. Jarden, Field Arithmetic, Ergeb. Math. Grenzgeb. (3) 11, Springer, Berlin, 1986.
  • \lccP. Gille, La R-équivalence sur les groupes algébriques réductifs définis sur un corps global, Inst. Hautes études Sci. Publ. Math. 86 (1997), 199–235.
  • ––––, $R$-équivalence sur les $G$-revêtements sur les corps locaux non archimédiens, J. Number Theory 91 (2001), 284–292.
  • \lccJ. Harris, B. Mazur, and R. Pandharipande, Hypersurfaces of low degree, Duke Math. J. 95 (1998), 125–160.
  • \lccW. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge Univ. Press, Cambridge, 1947.
  • \lccK. Kato and S. Saito, Unramified class field theory of arithmetical surfaces, Ann. of Math. (2) 118 (1983), 241–275.
  • \lccB. Kim and R. Pandharipande, "The connectedness of the moduli space of maps to homogeneous spaces" in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 187–201.
  • \lccJ. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • ––––, Rationally connected varieties over local fields, Ann. of Math. (2) 150 (1999), 357–367.
  • ––––, Fundamental groups of rationally connected varieties, Michigan Math. J. 48 (2000), 359–368.
  • ––––, Which are the simplest algebraic varieties?, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 409–433.
  • ––––, "Rationally connected varieties and fundamental groups" in Higher Dimensional Varieties and Rational Points (Budapest, 2001), ed. K. Böröczky Jr., J. Kollár, and T. Szamuely, Bolyai Soc. Math. Stud. 12, Springer, New York, 2003, 69–92.
  • ––––, letter to J. Starr, 2001.
  • \lccJ. Kollár, Y. Miyaoka, and S. Mori, Rationally connected varieties, J. Algebraic Geometry 1 (1992), 429–448.
  • \lccS. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827.
  • \lccD. Madore, Équivalence rationnelles sur les hypersurfaces cubiques sur les corps $p$-adiques, Manuscripta Math. 110 (2003), 171–185. \CMP1 962 532
  • \lccYu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Math. Library 4, North-Holland, Amsterdam, 1986.
  • \lccL. Moret-Bailly, R-équivalence simultanée de torseurs: un complément à l'article de P. Gille, J. Number Theory 91 (2001), 293–296.
  • ––––, Sur la R-équivalence de torseurs sous un groupe fini, J. Number Theory 99 (2003), 383–404. \CMP1 968 460
  • \lccS. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), 593–606.
  • \lccD. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204.
  • \lccJ.-P. Serre, Corps locaux, Publ. Inst. Math. Univ. Nancago 8, Actualités Sci. Indust. 1296, Hermann, Paris, 1962.
  • \lccT. Shioda and T. Katsura, On Fermat varieties, Tôhoku Math. J. 31 (1979), 97–115.
  • \lccH. P. F. Swinnerton-Dyer, "Universal equivalence for cubic surfaces over finite and local fields" in Symposia Mathematica, Vol. 24 (Rome, 1979), Academic Press, London, 1981, 111–143. La $R$-équivalence sur les tores, Ann. Sci. école Norm. Sup. (4) 10 (1977), 175–229.
  • \lccJ.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces, I, J. Reine Angew. Math. 373 (1987), 37–107.
  • ––––, Intersections of two quadrics and Châtelet surfaces, II, J. Reine Angew. Math. 374 (1987), 72–168.
  • \lccO. Debarre, Variétés rationnellement connexes, to appear in Séminaire Bourbaki 2001/02, exp. 905, \wwwwww-irma.u-strasbg.fr/~debarre
  • \lccH. Esnault, Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math. 151 (2003), 187–191. \CMP1 943 746
  • \lccW. Fulton and R. Pandharipande, "Notes on stable maps and quantum cohomology" in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, 1997, 45–96.
  • \lccM. Fried and M. Jarden, Field Arithmetic, Ergeb. Math. Grenzgeb. (3) 11, Springer, Berlin, 1986.
  • \lccP. Gille, La R-équivalence sur les groupes algébriques réductifs définis sur un corps global, Inst. Hautes études Sci. Publ. Math. 86 (1997), 199–235.
  • ––––, $R$-équivalence sur les $G$-revêtements sur les corps locaux non archimédiens, J. Number Theory 91 (2001), 284–292.
  • \lccJ. Harris, B. Mazur, and R. Pandharipande, Hypersurfaces of low degree, Duke Math. J. 95 (1998), 125–160.
  • \lccW. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge Univ. Press, Cambridge, 1947.
  • \lccK. Kato and S. Saito, Unramified class field theory of arithmetical surfaces, Ann. of Math. (2) 118 (1983), 241–275.
  • \lccB. Kim and R. Pandharipande, "The connectedness of the moduli space of maps to homogeneous spaces" in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 187–201.
  • \lccJ. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • ––––, Rationally connected varieties over local fields, Ann. of Math. (2) 150 (1999), 357–367.
  • ––––, Fundamental groups of rationally connected varieties, Michigan Math. J. 48 (2000), 359–368.
  • ––––, Which are the simplest algebraic varieties?, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 409–433.
  • ––––, "Rationally connected varieties and fundamental groups" in Higher Dimensional Varieties and Rational Points (Budapest, 2001), ed. K. Böröczky Jr., J. Kollár, and T. Szamuely, Bolyai Soc. Math. Stud. 12, Springer, New York, 2003, 69–92.
  • ––––, letter to J. Starr, 2001.
  • \lccJ. Kollár, Y. Miyaoka, and S. Mori, Rationally connected varieties, J. Algebraic Geometry 1 (1992), 429–448.
  • \lccS. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827.
  • \lccD. Madore, Équivalence rationnelles sur les hypersurfaces cubiques sur les corps $p$-adiques, Manuscripta Math. 110 (2003), 171–185. \CMP1 962 532
  • \lccYu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Math. Library 4, North-Holland, Amsterdam, 1986.
  • \lccL. Moret-Bailly, R-équivalence simultanée de torseurs: un complément à l'article de P. Gille, J. Number Theory 91 (2001), 293–296.
  • ––––, Sur la R-équivalence de torseurs sous un groupe fini, J. Number Theory 99 (2003), 383–404. \CMP1 968 460
  • \lccS. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), 593–606.
  • \lccD. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204.
  • \lccJ.-P. Serre, Corps locaux, Publ. Inst. Math. Univ. Nancago 8, Actualités Sci. Indust. 1296, Hermann, Paris, 1962.
  • \lccT. Shioda and T. Katsura, On Fermat varieties, Tôhoku Math. J. 31 (1979), 97–115.
  • \lccH. P. F. Swinnerton-Dyer, "Universal equivalence for cubic surfaces over finite and local fields" in Symposia Mathematica, Vol. 24 (Rome, 1979), Academic Press, London, 1981, 111–143.