= Solution
Write $F=\sum_C\min_{G\in|H|}\operatorname{mult}_C(G)\,C$. This is the <fixed part of a linear system>. The <movable part of a linear system> is $|M|$, with $M=H-F$; multiplication by the defining section of $F$ identifies its sections with those of $H$. In particular $|H|=F+|M|$, and no curve occurs in every member of $|M|$. The occurrence of $|F|$ in the definition request is understood as $|H|$.
For any integral curve $C$, choose a member $G\in|M|$ not containing $C$. Their <intersection number> is a sum of nonnegative local <intersection multiplicities>, so $M\cdot C=G\cdot C\geq0$. Therefore \b[the movable divisor $M$ is nef]. If $M=0$, the same assertion is immediate.
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