= 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> and
$$
\boxed{|D|=|D-C|+C.}
$$
Equivalently, multiplication by its defining section induces an isomorphism $H^0(X,\mathcal O_X(D-C))\xrightarrow{\sim}H^0(X,\mathcal O_X(D))$. The original PDF has this linear-system equality; the TeX loses the bars and reduces it to an uninformative divisor identity.
Back to article page
