Some reducible and irreducible Brill-Noether loci

We investigate limit linear series on chains of elliptic curves, giving a simple proof of a conjecture of Farkas stating the existence of curves with a theta-characteristic with a given number of sections for the expected range of genera. Using the additional structure afforded by considering limit linear series on chains of elliptic curves, we find examples of reducible Brill--Noether loci, admitting at least two components, with and without a theta-characteristic respectively. This allows us to display reducible Hilbert schemes for $r\ge 3$ and $d=g-1$. We also give examples of Brill--Noether loci with three components. On the positive side, we provide optimal bounds on the degree under which Brill--Noether loci are irreducible when $r=2$.