Rationall(upsilon) connected varieties over finite fields

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