A remark on the conjecture of Donagi-Morrison

TL;DR

This paper investigates the relationship between line bundles on K3 surfaces and their restrictions to curves, providing conditions under which certain linear systems are contained within others and comparing their Clifford indices.

Contribution

It proves that under specific genus and degree conditions, a line bundle on a K3 surface can be found that contains a given linear system on a curve and has a Clifford index not exceeding that of the system.

Findings

Existence of a line bundle N on X containing A on C under certain conditions.

Comparison of Clifford indices between N|C and A.

Conditions relating genus, degree, and linear systems on K3 surfaces.

Abstract

Let $X$ be a K3 surface, let $C$ be a smooth curve of genus $g$ on $X$, and let $A$ be a base point free and primitive line bundle $g_d^r$ on $C$ with $d\geq4$ and $r\geq\sqrt{\frac{d}{2}}$. In this paper, we prove that if $g>2d-3+(r-1)^2$, then there exists a line bundle $N$ on $X$ which is adapted to $|C|$ such that $|A|$ is contained in the linear system $|N\otimes\mathcal{O}_C|$, and ${\rm{Cliff}}(N\otimes\mathcal{O}_C)\leq {\rm{Cliff}}(A)$.