Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces

Abstract

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 $\operatorname{Cliff} (c_1(\mathcal{G} )_{|C})\leq \operatorname{Cliff} (A)$, 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 $\mathcal{O} _S(C)$.