  We explore some of the interplay between Brill-Noether subvarieties of the
moduli space SU_C(2,K) of rank 2 bundles with canonical determinant on a smooth
projective curve and 2\theta divisors, via the inclusion of the moduli space
into |2\theta|, singular along the Kummer variety. In particular we show that
the moduli space contains all the trisecants of the Kummer and deduce that
there are quadrisecant lines only if the curve is hyperelliptic; we show that
for generic curves of genus <6, though no higher, bundles with >2 sections are
cut out by \Gamma_00; and that for genus 4 this locus is precisely the
Donagi-Izadi nodal cubic threefold associated to the curve.
