Remarks on Ruled surfaces and rank two bundles with canonical determinant and 4 sections

Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let $B^k(2,K_C)$ be the Brill-Noether loci parametrizing classes of (semi)-stable vector bundles $E$ of rank two with canonical determinant over $C$ with $h^0(C,E)\ge k$. We show that $B^4(2,K_C)$ it has an irreducible component $\mathcal{B}$ of dimension $3g-13$ on a general curve $C$ of genus $g\ge 8$. Moreover, we show that for the general element $[E]$ of $\mathcal{B}$, $E$ fits in a exact sequence $0\to {\mathcal{O}}_C(D)\to E\to K_C(-D)\to 0$, with $D$ a general effective divisor of degree three, and the corresponding coboundary map $\mathrm{\partial }:H^0(C,K_C(-D))\to H^1(C,{\mathcal{O}}_C(D))$ has cokernel of dimension three.