A remark on the conjecture of Donagi-Morrison

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 $d\ge 4$ and $r\ge \sqrt{\frac{d}{2}}$. 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 $\mathrm{Cliff}(N\otimes {\mathcal{O}}_C)\le \mathrm{Cliff}(A)$.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:2310.19338