Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pencils and nets on curves arising from rank 1 sheaves on K3 surfaces   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 $\mathcal{O}_S(C)$.

9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int