Brill--Noether Generality of Curves and K3 Surfaces

Lazarsfeld proved Brill–Noether generality of any smooth curve in the linear system $\mathrm{\mid }H\mathrm{\mid }$ where $(X,H)$ is a polarized K3 surface with $\mathrm{P}\mathrm{i}\mathrm{c}(X)=\mathbb{Z}\cdot H$. Mukai introduced the notion of Brill–Noether generality for quasi-polarized K3 surfaces. We prove Brill–Noether generality of any smooth curve in the linear system $\mathrm{\mid }H\mathrm{\mid }$ where $(X,H)$ is a Brill–Noether general quasi-polarized K3 surface.