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

Part of: Surfaces and higher-dimensional varieties Cycles and subschemes

Published online by Cambridge University Press:  14 July 2015

V. Guletskiĭ  and
A. Tikhomirov
V. Guletskiĭ
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK, (vladimir.guletskii@liverpool.ac.uk)
A. Tikhomirov*
Affiliation:
Department of Mathematics, Yaroslavl State University, 108 Respublikanskaya Street, Yaroslavl 150000, Russia, (astikhomirov@mail.ru)
*
* Present address: Department of Mathematics, Higher School of Economics, 7 Vavilova Street, Moscow 117312, Russia, atikhomirov@hse.ru
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Abstract

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 and Πτ 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 𝓅3 associated 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.

Keywords

algebraic cyclesK3-surfacessymplectomorphismmixed Hodge structurescubic threefoldsintermediate Jacobian

MSC classification

Primary: 14C25: Algebraic cycles 14J28: $K3$ surfaces and Enriques surfaces 14J70: Hypersurfaces 14J30: $3$-folds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 377 - 392
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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