By a theorem of H. Clemens, the Griffiths group of codimension 2 cycles on a very general quintic threefold is of infinite rank. C. Voisin gave a proof of this result using infinitesimal methods. In this paper the author applies Voisin’s techniques to study the image of the regulator map R:CH2(S, 1) to H3D (S,Z(2)) on Bloch’s higher Chow group for a quartic surface S. It is known that the image of this map is nontrivial if S is very general by work of Voisin and C. Oliva (unpublished) and S. J.M¨uller-Stach [J. Algebraic Geom. 6 (1997), (M¨uller-Stach treated the case of quartic surfaces that contain a line). The author proves that the image of the regulator map R is not finitely generated if S is very general.
Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds.(English summary)Algebraic K-theory and its applications (Trieste, 1997), 370–402, World Sci. Publ., River Edge,NJ, 1999.
COLLINO, Alberto
1999-01-01
Abstract
By a theorem of H. Clemens, the Griffiths group of codimension 2 cycles on a very general quintic threefold is of infinite rank. C. Voisin gave a proof of this result using infinitesimal methods. In this paper the author applies Voisin’s techniques to study the image of the regulator map R:CH2(S, 1) to H3D (S,Z(2)) on Bloch’s higher Chow group for a quartic surface S. It is known that the image of this map is nontrivial if S is very general by work of Voisin and C. Oliva (unpublished) and S. J.M¨uller-Stach [J. Algebraic Geom. 6 (1997), (M¨uller-Stach treated the case of quartic surfaces that contain a line). The author proves that the image of the regulator map R is not finitely generated if S is very general.
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