Rank two bundles with canonical determinant and four sections

Let \(C\) be a smooth irreducible complex projective curve of genus \(g\) and let \(B^k(2,K_C)\) be the Brill–Noether locus 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 \(B^4(2,K_C)\) 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 into an exact sequence \(0\rightarrow {\mathcal {O}}_C(D)\rightarrow E\rightarrow K_C(-D)\rightarrow 0\) with \(D\) a general effective divisor of degree three, and the corresponding coboundary map \(\partial : H^0(C,K_C(-D))\rightarrow H^1(C,{\mathcal {O}}_C(D))\) has cokernel of dimension three.