NONEMPTINESS OF BRILL–NOETHER LOCI

Abstract

Let X be a non-singular algebraic curve of genus g. We prove that the Brill–Noether locus [Formula: see text] is non-empty if d=nd′+d′′ with 0<d′′< 2n, 1≤s≤g, d′≥(s-1)(s+g)/s, n≤d′′+(n-k)g, (d′′,k) ≠(n, n). These results hold for an arbitrary curve of genus ≥ 2, and allow us to construct a region in the associated "Brill–Noether (μ, λ)-map" of points for which the Brill–Noether loci are non-empty. Even for the generic case, the region so constructed extends beyond that defined by the so-called "Teixidor parallelograms". For hyperelliptic curves, the same methods give more extensive and precise results.