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

Abstract

Let C be a smooth projective algebraic curve of genus g, over the finite field Fq. A classical result of H. Martens states that the Brill-Noether locus of line bundles L in PicdC with deg L = d and h0(C,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 Fq is bounded above by Kgqd-2i+2, with an explicit constant Kg 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 - Kg′/q for the probability that a line bundle in Picg+1C(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. © 2018 World Scientific Publishing Company.