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
Full-text PDF

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.


References [Enhancements On Off] (What's this?)


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

DOI: https://doi.org/10.1090/S1056-3911-2014-00631-7
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.

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website