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Article

Algebraic Cycles on Quadric Sections of Cubics in ℙ4 under the Action of Symplectomorphisms

Proceedings of the Edinburgh Mathematical Society. 2016. Vol. 59. No. 02. P. 377-392.
Guletski ̆i V., Tikhomirov A. S.

Let τ be the involution changing the sign of two coordinates in ℙ4. We prove that τ induces the identity action on the second Chow group of the intersection of a τ-invariant cubic with a τ-invariant quadric hypersurface in ℙ4. Let lτ andΠτ be the one- and two-dimensional components of the fixed locus of the involution τ. We describe the generalized Prymian associated with the projection of a τ-invariant cubic 𝓵 ⊂ P4 from lτ onto Πτ in terms of the Prymians 𝓅2 and 𝓅3associated with the double covers of two irreducible components, of degree 2 and 3, respectively, of the reducible discriminant curve. This gives a precise description of the induced action of the involution τ on the continuous part of the Chow group CH2 (𝓵). The action on the subgroup corresponding to 𝓅3 is the identity, and the action on the subgroup corresponding to 𝓅2 is the multiplication by —1.