Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{ O}_S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld--Mukai bundles to prove that $A$ is cut out by the global sections of a rank 1 torsion-free sheaf $\mathcal{ G}$ on $S$. Furthermore, we show that $c_1(\mathcal{ G})$ with one exception is adapted to $\mathcal{ O}_S(C)$ and satisfies $\mathrm{ Cliff}(c_1(\mathcal{ G})_{ |C})\leq\mathrm{ Cliff}(A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case. In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{ G})_{ |C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{ O}_S(C)$.
Categories: math.AG