A remark on the conjecture of Donagi-Morrison

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 $\mathrm{\mid }C\mathrm{\mid }$ such that $\mathrm{\mid }A\mathrm{\mid }$ is contained in the linear system $\mathrm{\mid }N\otimes {\mathcal{O}}_C\mathrm{\mid }$, and $\mathrm{C}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{f}(N\otimes {\mathcal{O}}_C)\le \mathrm{C}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{f}(A)$.