The Donagi–Morrison conjecture for Brill–Noether series on K3 surfaces

Let $(S,H)$ be a polarized K3 surface and let $C\in|H|$ be a smooth irreducible curve of genus $g\ge 2$. Suppose $A$ is a complete basepoint-free $g^r_d$ on $C$ with $d\le g-1$ and $\rho(g,r,d)<0$. A line bundle $M\in\operatorname{Pic}(S)$ is adapted to $|H|$ if it has the properties required in the Donagi–Morrison formulation. Donagi–Morrison conjecture. There exists a line bundle $M\in\operatorname{Pic}(S)$ adapted to $|H|$ such that $|A|$ is contained in the restriction of $|M|$ to $C$ and

$$\gamma(M\otimes\mathcal{O}_C)\le\gamma(A).$$

Donagi and Morrison proved the assertion for complete basepoint-free Brill–Noether special pencils, and the stated version incorporates slight modifications due to Lelli-Chiesa. The general assertion in the stated range is presented as conjectural.