a general curve $[C]\in \cM_g$, we have that $\mathcal{BN}_C(d, 4)\neq \emptyset$ whenever $\beta_g(d, 4)\geq 0$.

Using the already mentioned connection between coherent systems $(E, V)$, where $E\in \cU_C(2, d)$ and $V\in G(4, H^0(C, E))$ on one side, and the non-vanishing of the cohomology group $K_{1, 1}(C, \mbox{det}(E))$ on the other, Theorem \[existence\] is implied by the following:

\[quad4\] For a general curve $[C]\in \cM_{2a+1}$, the locus of special linear series $$\mathfrak{Koszul}(C):=\{L\in W^4_{2a+4}(C): \mathrm{Sym}^2 H^0(C, L)\stackrel{\nu_2(L)}\longrightarrow H^0(C, L^{\otimes 2})\ \ \mbox{ is not injective}\}$$ has at least one component of dimension $2$, whose general element corresponds to a complete base point free linear series, which cannot be written as $L=A_1\otimes A_2$, with $A_1, A_2\in W^1_{a+2}(C)$.

Assuming Theorem \[quad4\], the corresponding vector bundle $E\in \mathcal{BN}_C(2a+4, 4)$ is constructed as a twist of a *Lazarsfeld* bundle on $C$. Precisely, for $L\in \mathfrak{Koszul}(C)$, we take $E:=M_{W}\otimes L$, where $W\in G(3, H^0(C, L))$ is a suitably chosen subspace such that $\mbox{Ker } \nu_2(L)\cap \bigl(W\otimes H^0(C, L)\bigr)\neq 0$. This method of constructing $E$ is the first instance of a general construction of vector bundles starting from non-trivial Koszul cohomology classes of small rank [@vB], [@AN]. We refer to Section 5 for details.

Next we turn to 