Monogonal linear system on a K3 surface ($D=mE+\Gamma ,m\ge 2$)

For a nef big divisor, the monogonal case is the fixed-section system $|D|=\Gamma +|mE|$, where $E$ is an elliptic pencil on a K3 surface and $\Gamma$ is a section. Thus $D\cdot E=1$. The fixed part persists even though $D$ is nef; the general divisor is reducible. Riemann–Roch theorem for algebraic surfaces gives $h^0(D)=m+1=h^0(mE)$, proving the equality of systems.