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We consider the involution changing the sign of two coordinates in4-dimensional projective space. The intersection S of invariant cubic andquadric hypersurfaces in P^4 is a K3-surface with the induced symplectomorphicaction on its second cohomology group. The Bloch-Beilinson conjecture predictsthat 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 theidentity action on CH^2 for K3-surfaces as above. Then we look at a smoothinvariant cubic hypersurface C in P^4 and project it from the 1-dimensionalonto the 2-dimensional linear spaces of the fixed locus of the involution. Thediscriminant curve splits into two components of degree 2 and 3, and thegeneralized Prymian can be described in terms of the Prymians P_2 and P_3associated to the double covers of these components. Such an approach givesprecise information about the induced action of the involution on thecontinuous part in the second Chow group CH^2(C) of the threefold C, as well ason the Hodge pieces of its third cohomology group. The action on the subgroupin CH^2(C) corresponding to the Prymian P_3 is the identity, and the action onthe subgroup corresponding to P_2 is the multiplication by -1.

Algebraic cycles on quadric sections of cubics in P^4 under the action of symplectomorphisms