  This partly expository paper investigates versions of the Tate conjecture on
the cycle map for varieties defined over finite fields with values in 'etale
cohomology with Z_\ell-coefficients. The bulk of the paper is an exposition of
a 1998 result of C. Schoen which shows that an integral version of the
conjecture holds for 1-cycles provided the usual conjecture is true for
divisors on surfaces. In a last section we then derive from Schoen's theorem
new results on the existence of degree one zero-cycles on varieties defined
over the function field of a smooth proper curve C over the algebraic closure
of a finite field. In particular, we show that if the variety in question is a
smooth projective complete intersection of dimension at least 3 and of degree
prime to the characteristic, then a zero-cycle of degree one exists if the Tate
conjecture is true for divisors on surfaces and the variety extends to a proper
fibration over C all of whose fibres possess a component of multiplicity one.
