complete intersection $K3$ surface. As a consequence, $\mathcal{BN}_C(2a+3, 4)\neq \emptyset$ and Mercat’s Conjecture fails for $C$.

We use [@K] Theorem 6.1 to construct a curve $C\subset S\subset \PP^4$, lying on a smooth complete intersection surface of type $(2, 3)$ such that $\mbox{Pic}(S)=\mathbb Z\cdot H\oplus \mathbb Z\cdot C$, where $H^2=6,\ H\cdot C=2a+3$ and $C^2=4a$. Since $h^1(C, \OO_C(1))\geq 2$, it follows that $\OO_C(1)$ contributes to $\mathrm{Cliff}(C)$, hence $\mathrm{Cliff}(C)\leq \mathrm{Cliff}(\OO_C(1))=2a-5$. We aim to show that $\mathrm{Cliff}(C)=a$, that is, $C$ has maximal possible Clifford index.

Assume by contradiction that $\mathrm{Cliff}(C)<a$, which means that $\mbox{Cliff}(C)$ is computed by a line bundle which comes from $S$. Note by direct calculation that $S$ carries no $(-2)$ curves, in particular $C$ has Clifford dimension $1$. We reason along the lines of [@F1] Theorem 3. Using [@GL] we infer that there exists a curve $D\subset S$, satisfying $$\begin{aligned}
\label{conditions}
  & & h^0(S, \OO_S(D))=h^0(C, \OO_C(D))\geq 2, \nonumber\\
 & & h^0(S, \OO_S(C-D))=h^0(C, K_C(-D))\geq 2,\\
 & & C\cdot D\leq g-1, \nonumber\end{aligned}$$ such that $\mathrm{Cliff}(C)=\mathrm{Cliff}(D\otimes \OO_C)=D\cdot C-D^2-2$. In particular, such a divisor $D \equiv mH + nC$, with $m,n \in \ZZ$ must verify the inequalities: $$\begin{array}{rl}
\mbox{(i)} & D\cdot H= 6m + dn >2 \\
\mbox{(ii)} &  md + (2n-1 )(g-1) \leq 0 