Subvarieties of $\mathcal{SU}_C(2)$ and $2\theta$-divisors in the Jacobian

We explore some of the interplay between Brill-Noether subvarieties of the moduli space ${\mathcal{SU}}_C(2,K)$ 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.