Duke Mathematical Journal
- Duke Math. J.
- Volume 120, Number 2 (2003), 251-267.
Rationally connected varieties over finite fields
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 $Y_P$.
Duke Math. J., Volume 120, Number 2 (2003), 251-267.
First available in Project Euclid: 16 April 2004
Permanent link to this document
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14G15: Finite ground fields 14J20: Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] 14M20: Rational and unirational varieties [See also 14E08]
Secondary: 14C15: (Equivariant) Chow groups and rings; motives 14G20: Local ground fields
Kollár, János; Szabó, Endre. Rationally connected varieties over finite fields. Duke Math. J. 120 (2003), no. 2, 251--267. doi:10.1215/S0012-7094-03-12022-0. https://projecteuclid.org/euclid.dmj/1082138584