  Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the
Brill-Noether locus $\bns $ is non-empty if $d= nd' +d'' $ with $0< d'' <2n$,
$1\le s\le g$, $d'\geq (s-1)(s+g)/s $, $n\leq d''+(n-k)g$, $(d'',k)\ne(n,n)$.
These results hold for an arbitrary curve of genus $\ge 2$, and allow us to
construct a region in the associated ``Brill-Noether $\pa$-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.
