On the Chow ring of a K3 surface
Authors:
Arnaud Beauville and Claire Voisin
Journal:
J. Algebraic Geom. 13 (2004), 417-426
DOI:
https://doi.org/10.1090/S1056-3911-04-00341-8
Published electronically:
January 5, 2004
MathSciNet review:
2047674
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Abstract |
References |
Additional Information
Abstract: We show that the Chow group of $0$-cycles on a K3 surface contains a class of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class $c_2$.
[B]B A. Beauville: Sur l’anneau de Chow d’une variété abélienne. Math. Annalen 273 (1986), 647–651.
[Bl]Bl S. Bloch: Some elementary theorems about algebraic cycles on Abelian varieties. Invent. Math. 37 (1976), 215–228.
[Bl-S]Bl-S S. Bloch, V. Srinivas: Remarks on correspondences and algebraic cycles. Amer. J. Math. 105 (1983), 1235–1253.
[C]C G. Ceresa: $C$ is not algebraically equivalent to $C^{-}$ in its Jacobian. Ann. of Math. 117 (1983), 285–291.
[G-S]G-S B. Gross, C. Schoen: The modified diagonal cycle on the triple product of a pointed curve. Ann. Inst. Fourier (Grenoble) 45 (1995), 649–679.
[M]M D. Mumford: Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1968), 195–204.
[M-M]M-M S. Mori, S. Mukai: Mumford’s theorem on curves on $K3$ surfaces. Algebraic Geometry (Tokyo/Kyoto 1982), LNM 1016, 351–352; Springer-Verlag (1983).
[R]R A. A. Rojtman: The torsion of the group of $0$-cycles modulo rational equivalence. Ann. of Math. 111 (1980), 553–569.
[S]S T. Shioda: On the Picard number of a Fermat surface. J. Fac. Sci. Univ. Tokyo 28 (1982), 725–734.
[SGA6]SGA6 Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Lecture Notes in Math. 225, Springer-Verlag, Berlin-New York (1971).
[B]B A. Beauville: Sur l’anneau de Chow d’une variété abélienne. Math. Annalen 273 (1986), 647–651.
[Bl]Bl S. Bloch: Some elementary theorems about algebraic cycles on Abelian varieties. Invent. Math. 37 (1976), 215–228.
[Bl-S]Bl-S S. Bloch, V. Srinivas: Remarks on correspondences and algebraic cycles. Amer. J. Math. 105 (1983), 1235–1253.
[C]C G. Ceresa: $C$ is not algebraically equivalent to $C^{-}$ in its Jacobian. Ann. of Math. 117 (1983), 285–291.
[G-S]G-S B. Gross, C. Schoen: The modified diagonal cycle on the triple product of a pointed curve. Ann. Inst. Fourier (Grenoble) 45 (1995), 649–679.
[M]M D. Mumford: Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1968), 195–204.
[M-M]M-M S. Mori, S. Mukai: Mumford’s theorem on curves on $K3$ surfaces. Algebraic Geometry (Tokyo/Kyoto 1982), LNM 1016, 351–352; Springer-Verlag (1983).
[R]R A. A. Rojtman: The torsion of the group of $0$-cycles modulo rational equivalence. Ann. of Math. 111 (1980), 553–569.
[S]S T. Shioda: On the Picard number of a Fermat surface. J. Fac. Sci. Univ. Tokyo 28 (1982), 725–734.
[SGA6]SGA6 Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Lecture Notes in Math. 225, Springer-Verlag, Berlin-New York (1971).
Additional Information
Arnaud Beauville
Affiliation:
Institut Universitaire de France & Laboratoire J.-A. Dieudonné (UMR 6621 du CNRS), Université de Nice, Parc Valrose, F-06108 Nice cedex 2, France
MR Author ID:
33175
Email:
beauville@math.unice.fr
Claire Voisin
Affiliation:
Institut de Mathématiques de Jussieu (UMR 7586 du CNRS), Case 247, 4 place Jussieu, F-75252 Paris cedex 05, France
MR Author ID:
237928
Email:
voisin@math.jussieu.fr
Received by editor(s):
November 21, 2001
Published electronically:
January 5, 2004
