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We consider the involution changing the sign of two coordinates in 4-dimensional projective space. The intersection S of invariant cubic and quadric hypersurfaces in P^4 is a K3-surface with the induced symplectomorphic action on its second cohomology group. The Bloch-Beilinson conjecture predicts that the induced action on the second Chow group CH^2(S) must be the identity. Generalizing the method developed by C. Voisin we non-conjecturally prove the identity action on CH^2 for K3-surfaces as above. Then we look at a smooth invariant cubic hypersurface C in P^4 and project it from the 1-dimensional onto the 2-dimensional linear spaces of the fixed locus of the involution. The discriminant curve splits into two components of degree 2 and 3, and the generalized Prymian can be described in terms of the Prymians P_2 and P_3 associated to the double covers of these components. Such an approach gives precise information about the induced action of the involution on the continuous part in the second Chow group CH^2(C) of the threefold C, as well as on the Hodge pieces of its third cohomology group. The action on the subgroup in CH^2(C) corresponding to the Prymian P_3 is the identity, and the action on the subgroup corresponding to P_2 is the multiplication by -1.
In this short note, we simply collect some known results about representing algebraic cycles by various kind of nice (e.g. smooth, local complete intersection, products of local complete intersection) algebraic cycles, up to rational equivalence. We
We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme o
We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz moti
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We establish the analogue of the Friedlander-Mazur conjecture for Tehs reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than th