Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On relations among $ 1$-cycles on cubic hypersurfaces

Author: Mingmin Shen
Journal: J. Algebraic Geom. 23 (2014), 539-569
Published electronically: January 23, 2014
MathSciNet review: 3205590
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Abstract: In this paper we give two explicit relations among $ 1$-cycles modulo rational equivalence on a smooth cubic hypersurface $ X$. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape's theorem that $ \mathrm {CH}_1(X)$ is always generated by lines and that it is isomorphic to $ \mathbb{Z}$ if the dimension of $ X$ is at least 5. Another application is to the intermediate jacobian of a cubic threefold $ X$. To be more precise, we show that the intermediate jacobian of $ X$ is naturally isomorphic to the Prym-Tjurin variety constructed from the curve parameterizing all lines meeting a given rational curve on $ X$. The incidence correspondences play an important role in this study. We also give a description of the Abel-Jacobi map for 1-cycles in this setting.

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Additional Information

Mingmin Shen
Affiliation: Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Address at time of publication: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Received by editor(s): May 21, 2011
Received by editor(s) in revised form: May 29, 2012
Published electronically: January 23, 2014
Article copyright: © Copyright 2014 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.