Elliptic pencil on a K3 surface ($E^2=0,|E|\cong {\mathbb{P}}^1$)

A primitive effective nef class of square zero on a complex K3 surface defines a basepoint-free pencil with connected genus-one fibres. Its general fibre is a smooth elliptic curve. A section is a smooth rational curve $\Gamma$ with $E\cdot \Gamma =1$ and ${\Gamma }^2=-2$.

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.