### Abstract

Let X be a geometrically rational (or, more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough, then X contains many rational curves defined over K. As a consequence we prove that R-equivalence is trivial on X if K is large enough. These results imply the following conjecture of J.-L. Colliot-Thélène: Let Y be a rationally connected variety over a number field F. For a prime P, let Y_{P} denote the corresponding variety over the local field F_{P}. Then, for almost all primes P, the Chow group of 0-cycles on Y_{P} is trivial and R-equivalence is trivial on Y_{P}.

Original language | English (US) |
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Pages (from-to) | 251-267 |

Number of pages | 17 |

Journal | Duke Mathematical Journal |

Volume | 120 |

Issue number | 2 |

DOIs | |

State | Published - Nov 1 2003 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Kollár, J., & Szabó, E. (2003). Rationally connected varieties over finite fields.

*Duke Mathematical Journal*,*120*(2), 251-267. https://doi.org/10.1215/S0012-7094-03-12022-0