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

Nils Henry Rasmussen

• Let S be a K3 surface, C a smooth curve on S with OS(C) ample, and A a base-point free g2d 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 c1(G) with one exception is adapted to OS(C) and satisfies Cliff(c1(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 g1d case.

  In the final section, we give an example of the mentioned exception where h0(C,c1(G)|C) is dependent on the curve C in its linear system, thereby failing to be adapted to OS(C).