Nonemptiness of Brill-Noether loci

Let X be a non-singular algebraic curve of genus g. We prove that the Brill -Noether locus W-n,d(sk-1) is non-empty if d = nd' + d" with 0 < d" < 2n, 1 less than or equal to s less than or equal to g, d' greater than or equal to (s - 1)(s + g)/s, n less than or equal to d" + (n - k)g, (d", k) not equ al (n, n). These results hold for an arbitrary curve of genus greater than or equal to 2, and allow us to construct a region in the associated "Brill- Noether (mu, lambda)-map" of points for which the Brill-Noether loci are no n-empty. Even for the generic case, the region so constructed extends beyon d that defined by the so-called "Teixidor parallelograms". For hyperellipti c curves, the same methods give more extensive and precise results.