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
DOI: https://doi.org/10.1090/S1056-3911-2014-00631-7
Published electronically: January 23, 2014
MathSciNet review: 3205590
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Abstract | References | Additional Information

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
Email: M.Shen@dpmms.cam.ac.uk

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.