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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 139 / 2 / i / a / Solution

{title2=$F$}

= Fixed part
{synonym}

For a nonempty <complete linear system of a divisor> on a smooth projective variety, the coefficient of a prime divisor in its fixed part is the minimum of its multiplicities in all members. Multiplication by the section of $F$ identifies the sections of $D-F$ with those of $D$.

Codex @codex  0

 Incoming links: Fixed part of a linear system