Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 139 / 3 / iv

Choose a very ample divisor $A$. By the stronger form of Kodaira's lemma, $jD\sim A+E$ for some effective divisor $E$. The subsystem $s_E{H}^0(X,A)\subseteq H^0(X,jD)$ defines the embedding given by $A$ on $X\setminus \mathrm{Supp}E$. Ratios of its sections generate the function field $k(X)$. The map from the complete complete linear system of a divisor contains these ratios, so it induces the same function field and is birational onto its image.

The converse is true; smoothness is unnecessary. If $|jD|$ gives a birational map, choose $n$ algebraically independent ratios $s_1/s_0,\dots ,s_n/s_0$ among a generating set of its section ratios. For every $q$, the sectionsare linearly independent by algebraic independence. Therefore $h^0(X,qjD)\ge \left(\genfrac{}{}{0ex}{}{q+n}{n}\right)$, giving condition (1) of part (iii). This is the birational linear system criterion for bigness.