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

Write $F={\sum }_C{\min }_{G\in |H|}{\mathrm{mult}}_C(G)C$. 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 $M\cdot C=G\cdot C\ge 0$. Therefore the movable divisor $M$ is nef. If $M=0$, the same assertion is immediate.

The relevant invariant of a divisor is its Iitaka dimension, also called its Kodaira dimension. If two linearly independent global sections $s,t$ occur in $H^0(X,nH)$, their ratio is a nonconstant rational function. The sections $s^q,s^{q-1}t,\dots ,t^q$ are linearly independent, because a nonconstant rational function over an algebraically closed field is transcendental over that field. Thus some complete linear system of a divisor has positive-dimensional image, contradicting $\kappa (H)=0$.

Consequently $h^0(X,nH)=1$ for every $n>0$; the nonemptiness of $|H|$ supplies a nonzero section in every multiple. Its unique divisor is entirely fixed. Hence every movable part $M_n$ is zero.