Skip to Main Content
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
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?)

  • Allen B. Altman and Steven L. Kleiman, Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), no. 1, 3–47. MR 569043
  • Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 472843
  • Arnaud Beauville, Les singularités du diviseur $\Theta $ de la jacobienne intermédiaire de l’hypersurface cubique dans ${\bf P}^{4}$, Algebraic threefolds (Varenna, 1981) Lecture Notes in Math., vol. 947, Springer, Berlin-New York, 1982, pp. 190–208 (French). MR 672617
  • 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
  • F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 539–545 (French). MR 1191735
  • C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI
  • 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
  • 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, DOI
  • V. Kanev, Principal polarizations of Prym-Tjurin varieties, Compositio Math. 64 (1987), no. 3, 243–270. MR 918413
  • 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
  • János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
  • Arthur Mattuck, Secant bundles on symmetric products, Amer. J. Math. 87 (1965), 779–797. MR 199196, DOI
  • 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
  • J. P. Murre, Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math. 25 (1972), 161–206. MR 352088
  • 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
  • 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
  • Kapil H. Paranjape, Cohomological and cycle-theoretic connectivity, Ann. of Math. (2) 139 (1994), no. 3, 641–660. MR 1283872, DOI
  • A. N. Tjurin, Five lectures on three-dimensional varieties, Uspehi Mat. Nauk 27 (1972), no. 5, (167), 3–50 (Russian). MR 0412196

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.