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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 𝓅 3 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 𝓅 3 is the identity, and the action on the subgroup corresponding to 𝓅 2 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

Information

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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