Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 134 / 2 / i / Solution

On a smooth algebraic surface, distinct integral curves have nonnegative intersection numbers, because their intersection multiplicities are nonnegative. For any member $E\in |D|$ of the complete linear system of a divisor, $E\cdot C=D\cdot C<0$. Hence $C$ must be a component of $E$; otherwise the intersection would be nonnegative.

Subtracting one copy of $C$ from every effective member gives precisely the effective members linearly equivalent to $D-C$. Conversely adding $C$ gives a member of $|D|$. Thus $C$ is a fixed component andEquivalently, multiplication by its defining section induces an isomorphism $H^0(X,{\mathcal{O}}_X(D-C))\stackrel{\sim }{\to }{H}^0(X,{\mathcal{O}}_X(D))$. The original PDF has this linear-system equality; the TeX loses the bars and reduces it to an uninformative divisor identity.