= Monogonal linear system on a K3 surface
{title2=$D=mE+\Gamma,\quad m\ge2$}
= Monogonal linear systems on a K3 surface
{synonym}
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.
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