for a general curve $[C]\in \cM_g$, the variety $\mathcal{BN}_C(d, k)$ is non-empty precisely when $\beta_g(d, k)\geq 0$, is false, and there are few uniform statements concerning the geometry of $\mathcal{BN}_C(d, k)$. A remarkable exception to such erratic behaviour is the highly interesting case of rank $2$ vector bundles with canonical determinant, which is clarified in [@T3].

To a bundle $E\in \mathcal{SU}_C(2, L)$ with $\mbox{det}(E)=L\in \mbox{Pic}(C)$ and $h^0(C, E)=p+3\geq 4$, following a construction introduced in [@V3] and developed in [@AN], one associates a non-trivial Koszul class $[\zeta(E)]\in K_{p, 1}(C, L)$. In this way, one establishes a dictionary between rank $2$ *Brill-Noether theory* and the *Koszul geometry* of $C$. For $p=1$, this procedure specializes to a more classical construction [@BV], [@M2], [@GMN], that assigns to a vector bundle $E\in \mathcal{SU}_C(2, L)$ with $h^0(C, E)=4$, a quadric $Q_E\in \mbox{Sym}^2 H^0(C, L)$ of rank at most $6$, containing the image $\phi_L(C)$ of $C$ under the map induced by $|L|$.

The starting point of our investigation was an attempt to translate, via this dictionary, various syzygetic results for curves in the style of [@AF], [@F3], into dimensionality problems for $\mathcal{BN}_C(d, k)$. For $k\leq 3$ and a general $[C]\in \cM_g$, the Brill-Noether locus $\mathcal{BN}_C(d, k)$ is 