Mercat’s generalization of Clifford’s inequality. For a semistable vector bundle $E$ of rank $2$ on $C$ and slope $\mu(E)$, Mercat [@Me] made an interesting prediction concerning its number of sections in terms of the Clifford index of the curve: $$\label{mercat1}
\mbox{If } \mathrm{Cliff}(C) + 2 \leq \mu(E) \leq g-1, \ \mbox{then }\
h^0(C, E) \leq 2+\mu(E) -\mathrm{Cliff}(C).$$ $$\label{mercat2}
\ \ \ \mbox{ If } \  1\leq \mu(E)\leq \mathrm{Cliff}(C)+2, \mbox{ then }\
h^0(C, E)\leq 2+\frac{1}{\mathrm{Cliff(C)+1}}\bigl(\mathrm{deg}(E)-2\bigr).$$

The conjecture is inspired by the case when $E$ can be written as an extension $$0\rightarrow A\rightarrow E\rightarrow A'\rightarrow 0,$$ where both line bundles $A, A'$ contribute to $\mathrm{Cliff}(C)$, in which case, (\[mercat1\]) is an immediate consequence of Clifford’s inequality applied to both $A$ and $A'$. For extensions of Clifford type inequalities to higher rank vector bundles and additional background, see [@LN].

We provide a counterexample to Mercat’s Conjecture when $h^0(C, E)=4$, which was the simplest case when the answer was unknown:

\[countermercat\] For each integer $a\geq 5$, there exist curves $[C]\in \cM_{2a+1}$ having maximal Clifford index $\mathrm{Cliff}(C)=a$, such that $\mathcal{BN}_C(2a+3, 4)\neq \emptyset.$ In particular Mercat’s Conjecture (\[mercat2\]) fails for $C$.

The counterexamples to Mercat’s Conjecture (also for $g=2a$, where $a\geq 6$, see Theorem \[K3surf2\]), are 