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

TL;DR

This paper establishes upper bounds on the number of rational line bundles with certain properties over finite fields, providing explicit constants and applications to probabilistic statements about base point freeness.

Contribution

It introduces a new upper bound for the count of rational line bundles with specified sections, independent of Martens' classical theorem, with explicit constants and applications.

Findings

Bound on the number of rational line bundles with h^0(L) >= i

Explicit exponential growth constant K_g for the bounds

Probability estimate for base point freeness of line bundles

Abstract

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.