Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T10:44:53.060Z Has data issue: false hasContentIssue false
  • English
  • Français

One-cycles on rationally connected varieties

Part of: Cycles and subschemes Special varieties

Published online by Cambridge University Press:  10 March 2014

Zhiyu Tian  and
Hong R. Zong
Zhiyu Tian
Affiliation:
Department of Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA email [email protected]
Hong R. Zong
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

Keywords

rationally connected varietiesalgebraic cycles

MSC classification

Secondary: 14M22: Rationally connected varieties 14C25: Algebraic cycles
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 3 , March 2014 , pp. 396 - 408
Copyright
© The Author(s) 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bloch, S. and Ogus, O., Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Supér (4) 7 (1974), 181–201.Google Scholar
Behrend, K. and Manin, Yu., Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), 1–60.Google Scholar
de Jong, A. J. and Starr, J., Every rationally connected variety over the function field of a curve has a rational point, Amer. J. Math. 125 (2003), 567–580.Google Scholar
Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75–109.Google Scholar
Floris, E., Fundamental divisors on Fano varieties of index n - 3, Geom. Dedicata 162 (2013), 1–7.Google Scholar
Fulton, W., Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542–575.Google Scholar
Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, in Notes on stable maps and quantum cohomology. Algebraic geometry Santa Cruz 1995, Part 2, Proceedings of Symposia in Pure Mathematics, vol. 62 (American Mathematical Society, Providence, RI, 1997), 45–96.Google Scholar
Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), 57–67.Google Scholar
Hartshorne, R., Complete Intersections and Connectedness, Amer. J. Math. 84 (1962), 497–508.CrossRefGoogle Scholar
Höring, A and Voisin, C, Anticanonical divisors and curve classes on Fano manifolds, Pure Appl. Math. Q. 7 (2011), 1371–1393; Special Issue: In memory of Eckart Viehweg.CrossRefGoogle Scholar
Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32 (Springer, Berlin, 1996).Google Scholar
Kollár, J., Holomorphic and pseudo-holomorphic curves on rationally connected varieties, Port. Math. 67 (2010), 155–179.Google Scholar
Voisin, C., On integral Hodge classes on uniruled and Calabi–Yau threefolds, in Moduli spaces and arithmetic geometry, Advanced Studies in Pure Mathematics, vol. 45 (World Scientific, 2006), 43–73.Google Scholar
Voisin, C., Remarks on curve classes on rationally connected varieties, in A celebration of algebraic geometry, Clay Mathematics Proceedings, vol. 18 (CMI/AMS publication, 2013), 591–599.Google Scholar
Zhu, Y., Fano hypersurfaces in positive characteristic, Preprint (2011), arXiv:1111.2964.Google Scholar
Zong, H. R., Curve classes on rationally connected varieties, Preprint (2012), arXiv:1207.0575.Google Scholar
Zong, H. R., On the space of conics on complete intersections, Preprint (2012), arXiv:1211.1946.Google Scholar