Abstract
In this paper we study the algebraic structure of the Tautological ring of a Jacobian: by the use of hard-Lefschetz-primitive classes we construct convenient generators that allow us to list and describe all the possible structures that may occur (the explicit list is given forg ≤ 9 and for a few special curves).
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References
E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris,Geometry of Algebraic Curves, I, Springer-Verlag, New York, 1985.
A. Beauville, Quelques remarques sur la tranformation de Fourier dans l’anneau de Chow d’une variété abélienne,Algebraic Geometry (Tokyo/Kyto, 1982), 238–260, Lecture Notes in Math.1016, Springer, Berlin, 1983.
A. Beauville, Sur l’anneau de Chow d’une variété abélienne,Math. Ann. 273 (1986), 647–651.
A. Beauville, Algebraic cycles on Jacobian varieties,Compositio Math. 140 (2004), 683–688.
G. Ceresa,C is not algebraically equivalent toC − in its Jacobian,Ann. of Math. (2)117 (1983), 285–291.
A. Collino, Poincaré’s formulas and hyperelliptic curves,Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 109 (1975), 89–101.
E. Colombo and B. van Geemen, Note on curves in a Jacobian,Compositio Math. 88 (1993), 333–353.
Ch. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform,J. Reine Angew. Math. 422 (1991), 201–219.
N. Fackhruddin, Algebraic cycles on generic abelian varieties,Compositio Math. 100 (1996), 101–119.
W. Fulton,Intersection Theory, Ergebnisse de Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1984.
M. Green, J. Murre, and C. Voisin,Algebraic Cycles and Hodge Theory, Lecture Notes in Mathematics1594, Springer-Verlag, Berlin, 1994.
F. Herbaut,Algebraic cycles on the Jacobian of a curve with a g d r, arXiv:math AG/0606140.
A. Ikeda, Algebraic cycles and infinitesima invariants on Jacobian varieties,J. Algebraic Geom. 12 (2003), 573–603.
K. Künnemann, On the Chow motive of an abelian scheme,Motives (Seatle, WA, 1991), 189–205, Proc. Sympos. Pure Math.55, Amer. Math. Soc., Providence, RI, 1994.
K. Künnemann, A Lefschetz decomposition for Chow motives of abelian schemes,Invent. Math. 113 (1993), 85–102.
A. Kouvidakis and G. Van Der Geer,Cycles relations on Jacobian varieties, arXiv:math AG/0608606.
G. Marini, Algebraic cycles on abelian varieties and their decomposition,Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat (8),7 (2004), 231–240.
S. Mukai, Duality betweenD(X) and\(D(\hat X)\) with its applications to Picard sheaves,Nagoya Math. J. 81 (1981), 153–175.
A. Polishchuk, Universal algebraic equivalences between tautological cycles on Jacobians of curves,Math. Z. 251 (2005), 875–897.
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Marini, G. Tautological cycles on Jacobian varieties. Collect. Math. 59, 167–190 (2008). https://doi.org/10.1007/BF03191366
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DOI: https://doi.org/10.1007/BF03191366