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

TL;DR

This paper proves a conjecture relating to the structure of certain linear systems on curves on K3 surfaces using Lazarsfeld–Mukai bundles, and explores exceptions where the expected properties do not hold.

Contribution

It establishes a link between base-point free g^2_d on curves and rank 1 sheaves on K3 surfaces, confirming a conjecture by Donagi and Morrison.

Findings

Confirmed the Donagi–Morrison conjecture for g^2_d linear systems.

Provided a new proof for the g^1_d case using Lazarsfeld–Mukai bundles.

Identified an exception where the sheaf's properties depend on the specific curve.

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