Upper bounds for some Brill-Noether loci over a finite field

Let C be a smooth projective algebraic curve of genus g over the finite field F_q. A classical result of H. Martens states that the Brill-Noether locus of line bundles L in Pic^d C with deg L = d and h^0(L) >= i is of dimension at most d-2i+2, under conditions that hold when such an L is both effective and special. We show that the number of such L that are rational over F_q is bounded above by K_g q^(d-2i+2), with an explicit constant K_g that grows exponentially with g. Our proof uses the Weil estimates for function fields, and is independent of Martens' theorem. We apply this bound to give a precise lower bound of the form 1 - K'_g/q for the probability that a line bundle in (Pic^(g+1) C)(F_q) is base point free. This gives an effective version over finite fields of the usual statement that a general line bundle of degree g+1 is base point free. This is applicable to the author's work on fast Jacobian group arithmetic for typical divisors on curves.