\mathrm{Pic}(S)$ such that $D^2=0$. By direct calculation, either $D\equiv C-D$, in which case $\mbox{Cliff}(\OO_C(D))=D\cdot C-D^2-2=2a-5\geq a$, or else, $D\equiv 2a H-3C$, hence $D\cdot C>g-1$, and $D$ cannot compute $\mathrm{Cliff}(C)$.

For genus $g=2a$, we have an analogous result in a similar range. We skip details:

\[K3surf2\] For $a\geq 6$, there exist smooth curves $C\subset \PP^4$ with $\mathrm{deg}(C)=2a+1$, $g(C)=2a$ and maximal Clifford index $\mathrm{Cliff}(C)=a-1$, such that $C$ is contained in a smooth $(2, 3)$ complete intersection $K3$ surface. It follows that $\mathcal{BN}_C(2a+1, 4)\neq \emptyset$.

It is important to realize that although (\[mercat2\]) (and very probably prediction (\[mercat1\]) as well), fail for certain Brill-Noether general curves, we still expect both Mercat conjectures to be valid for the generic curve. Theorem \[K3surf1\] should be interpreted as stating that the failure locus of (\[mercat2\]) is *not* a Brill-Noether locus in the classical sense, but rather a *Koszul subvariety* on $\cM_g$ in the style of [@F3], [@F5]. Precisely, the locus in $\cM_{2a+1}-\cM_{2a+1, a+1}^1$ where inequality (\[mercat2\]) does not hold, can be described as $$\mathfrak{Syz}_{g, 2a+3}^4:=\{[C]\in \cM_{2a+1}: \exists L\in W^4_{2a+3}(C)\ \mbox{ such that }\ K_{1, 1}(C, L)\neq 0\}.$$ This is a virtual Koszul divisor. Using the terminology of Section 2, a point $[C]$ lies in $\mathfrak{Syz}_{g, 2a+3}^4$ 