Pencils and nets on curves arising from rank 1 sheaves on K3 surfaces - Rasmussen, Nils Henry (MATHEMATICA SCANDINAVICA, 2018)
Let S be a K3 surface, C a smooth curve on S with 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 G on S. Furthermore, we show that c_1(G ) with one exception is adapted to O _S(C) and satisfies Cliff (c_1(G )_|C)≤ 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(G )_|C) is dependent on the curve C in its linear system, thereby failing to be adapted to O _S(C).