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Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles

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

Published online by Cambridge University Press:  18 December 2023

François Charles ,
Giovanni Mongardi  and
Gianluca Pacienza
François Charles
Affiliation:
DMA – École normale superiéure, 45 rue d'Ulm, 75230 Paris Cedex 05, France francois.charles@math.u-psud.fr
Giovanni Mongardi
Affiliation:
Dipartimento di Matematica, Alma Mater studiorum Universitá di Bologna, Piazza di Porta San Donato 5, Bologna 40126, Italy giovanni.mongardi2@unibo.it
Gianluca Pacienza
Affiliation:
Université de Lorraine, CNRS, IECL, F-54000 Nancy, France gianluca.pacienza@univ-lorraine.fr
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Abstract

We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic symplectic variety of $K3^{[n]}$-type that do not contain such a uniruled divisor. As an application, we provide a generalization of a result due to Beauville–Voisin on the Chow group of $0$-cycles on such varieties.

Keywords

rational curvesirreducible symplectic varietiesChow groups

MSC classification

Primary: 14C99: None of the above, but in this section 14J28: $K3$ surfaces and Enriques surfaces
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 2 , February 2024 , pp. 288 - 316
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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