in the dual Grassmannian $G\bigl(2, H^0(C, E)\bigr)$. Since $E$ contains no sub-pencils, it follows that $\PP(\mbox{Ker } \lambda)\cap Q^{\vee}=\emptyset$, that is, no hyperplane $H$ $$\Lambda\subset H\subset \PP$$ is tangent to $Q$. But this clearly implies that $\mbox{res}_C(Q)\neq 0$, for otherwise it would imply that $\Lambda \subset \mbox{Sing}(Q)$. This is impossible based on dimension reasons. Since every quadric containing $G(2, 5)\subset \PP^9$ is a Plücker quadric this finishes the proof.

We discuss how Proposition \[restrquadrics\] can be applied to study Mercat’s Conjecture. When $h^0(C, E)=4$, inequality (\[mercat1\]) is vacuous for curves of maximal Clifford index, while (\[mercat2\]) breaks into two vanishing statements depending on the parity of $g$:

\[mc2\] For $[C]\in \cM_{2a+1}$ with $\mathrm{Cliff}(C)=a$, is it true that $\mathcal{BN}_C(2a+3, 4)=\emptyset$? For a curve $[C]\in \cM_{2a}$ with $\mathrm{Cliff}(C)=a-1$, is it true that $\mathcal{BN}_C(2a+1, 4)=\emptyset$?

The answer to both these questions is negative. Using the surjectivity of the period map for $K3$ surfaces in the style of [@F1], [@K], we construct curves of maximal gonality and prescribed degree and genus, lying on $K3$ surfaces in $\PP^4$.

\[K3surf1\] For each integer $a\geq 5$, there exist smooth curves $C\subset \PP^4$ with $\mathrm{deg}(C)=2a+3$, $g(C)=2a+1$ and maximal Clifford index $\mathrm{Cliff}(C)=a$, such that $C$ lies on a smooth 