[Le motif de la surface des droites]
Dans cette note, nous prouvons que le motif de Chow de la surface des droites d'une hypersurface cubique lisse dans est de dimension finie dans le sens de Kimura. Ceci constitue un exemple d'une variété qui n'est pas domineé par un produit de courbes, mais dont le motif de Chow est de type abélien.
In this short note, we prove that the Chow motive of the Fano surface of lines on the smooth cubic threefold is finite-dimensional in the sense of Kimura. This gives an example of a variety not dominated by a product of curves whose Chow motive is of Abelian type.
Accepté le :
Publié le :
@article{CRMATH_2016__354_9_925_0,
author = {Humberto Diaz},
title = {The motive of the {Fano} surface of lines},
journal = {Comptes Rendus. Math\'ematique},
pages = {925--930},
publisher = {Elsevier},
volume = {354},
number = {9},
year = {2016},
doi = {10.1016/j.crma.2016.07.003},
language = {en},
}
Humberto Diaz. The motive of the Fano surface of lines. Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 925-930. doi : 10.1016/j.crma.2016.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.07.003/
[1] Les singularités du diviseur Θ de la jacobienne intermédiaire de l'hypersurface cubique dans , Varenna, 1981 (Lecture Notes in Math.), Volume vol. 947, Springer, Berlin, New York (1982), pp. 190-208
[2] Sur l'anneau de Chow d'une variété abélienne, Math. Ann., Volume 273 (1986), pp. 647-651
[3] Lectures on the Theory of Algebraic Cycles, Duke University Mathematics Series, vol. IV, Mathematics Department, Duke University, Durham, NC, USA, 1980
[4] The fundamental group of the Fano surface I, II, Varenna, 1981 (Lecture Notes in Math.), Volume vol. 947, Springer, Berlin (1982), pp. 209-218 (219–220)
[5] Motivic decomposition of Abelian schemes and the Fourier transform, Compos. Math., Volume 88 (1993) no. 3, pp. 333-353
[6] Intersection Theory, Ergebnisse, 3. Folge, vol. 2, Springer Verlag, 1984
[7] Motives and representability of algebraic cycles on threefolds over a field, J. Algebraic Geom., Volume 21 (2012) no. 2, pp. 347-373
[8] Motivic sheaves and filtrations on Chow groups (U. Jannsen; S. Kleiman; J.-P. Serre, eds.), Motives, Proceedings of Symposia in Pure Mathematics, Part 1, vol. 55, American Mathematical Society, Providence, RI, USA, 1994, pp. 245-302
[9] B. Kahn, Letter to the author, 2015.
[10] Chow groups are finite-dimensional, in some sense, Math. Ann., Volume 331 (2005) no. 1, pp. 173-201
[11] The standard conjectures (U. Jannsen; S. Kleiman; J.-P. Serre, eds.), Motives, Proceedings of Symposia in Pure Mathematics, Part 1, vol. 55, American Mathematical Society, Providence, RI, USA, 1994, pp. 3-20
[12] R. Laterveer, A remark on the motive of the Fano variety of lines of a cubic, preprint.
[13] On the motive of an algebraic surface, J. Reine Angew. Math., Volume 409 (1990), pp. 190-204
[14] Construction of Hilbert and Quot Schemes | arXiv
[15] On the Tate conjecture for the Fano surfaces of cubic threefolds | arXiv
[16] Theta divisors with curve summands and the Schottky problem | arXiv
[17] Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Princeton University Press, Princeton, NJ, USA, 2014
Cité par Sources :
Commentaires - Politique