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.
References
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References
- Allen B. Altman and Steven L. Kleiman, Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), no. 1, 3–47. MR 0569043 (58 \#27967)
- Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 0472843 (57 \#12532)
- Arnaud Beauville, Les singularités du diviseur $\Theta$ de la jacobienne intermédiaire de l’hypersurface cubique dans ${\mathbb P}^{4}$, Algebraic threefolds (Varenna, 1981) Lecture Notes in Math., vol. 947, Springer, Berlin, 1982, pp. 190–208 (French). MR 672617 (84c:14030)
- S. Bloch and J. P. Murre, On the Chow group of certain types of Fano threefolds, Compositio Math. 39 (1979), no. 1, 47–105. MR 539001 (80m:14025)
- F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 539–545 (French). MR 1191735 (93k:14050)
- C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 0302652 (46 \#1796)
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- Joe Harris, Mike Roth, and Jason Starr, Curves of small degree on cubic threefolds, Rocky Mountain J. Math. 35 (2005), no. 3, 761–817. MR 2150309 (2007a:14011), DOI https://doi.org/10.1216/rmjm/1181069707
- V. Kanev, Principal polarizations of Prym-Tjurin varieties, Compositio Math. 64 (1987), no. 3, 243–270. MR 918413 (88j:14038)
- János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
- János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625 (93i:14014)
- Arthur Mattuck, Secant bundles on symmetric products, Amer. J. Math. 87 (1965), 779–797. MR 0199196 (33 \#7345)
- David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510 (52 \#415)
- J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math. 25 (1972), 161–206. MR 0352088 (50 \#4576a)
- J. P. Murre, Some results on cubic threefolds, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 140–160. Lecture Notes in Math., Vol. 412. MR 0374145 (51 \#10345)
- Jacob P. Murre, Un résultat en théorie des cycles algébriques de codimension deux, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 23, 981–984 (French, with English summary). MR 777590 (86c:14004)
- Kapil H. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139 (1994), no. 3, 641–660. MR 1283872 (95g:14008), DOI https://doi.org/10.2307/2118574
- A. N. Tjurin, Five lectures on three-dimensional varieties, Uspehi Mat. Nauk 27 (1972), no. 5, (167), 3–50 (Russian). MR 0412196 (54 \#323)
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.
