$\mbox{gon}(C)=7$. After counting dimensions, we observe that there exists $X\in |I_{C/\PP^4}(3)|$, which does not contain $Q$, and such that $S:=Q\cap X\subset \PP^4$ is a smooth $K3$ surface. By direct calculation, we check that $h^0(S, \OO_S(H-C))\geq 2$ and $(H-C)^2=0$, that is, $S$ is an elliptic $K3$ surface. This contradicts the main result of [@M1], where it has been shown that a general curve of genus $11$ lies on a single $K3$ surface of degree $20$, which moreover is general in its moduli space, in particular it has Picard number one. $\Box$

We next turn to the case of globally generated vector bundles $E$ with $h^0(C, E)=5$ having no sub-pencils. We set as usual $L:=\mbox{det}(E)$ and then $h^0(C, L)\geq 7$.

\[5sections\] For a general curve $[C]\in \cM_{2a+1}$, Mercat’s Conjecture holds for vector bundles with $5$ sections, if and only if $\mathcal{BN}_C(2a+5, 5)=\emptyset$. Similarly, for even genus, Mercat’s Conjecture for $h^0(C, E)=5$ holds in the case of a general curve $[C]\in \cM_{2a}$, if and only if $\mathcal{BN}_C(2a+3, 5)=\emptyset$.

Via diagram (\[diag\]), we use the existence of the Plücker quadrics in the ideal of the curve $\phi_L(C)$ embedded by the determinant line bundle, to confirm (\[mercat1\]) in bounded genus: *Proof of Theorem \[mercatcinci\]:* We 