= Solution
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 $s_EH^0(X,A)\subseteq H^0(X,jD)$ defines the embedding given by $A$ on $X\setminus\operatorname{Supp}E$. 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 \b[birational onto its image].
\b[The converse is true; smoothness is unnecessary.] If $|jD|$ gives a birational map, choose $n$ algebraically independent ratios $s_1/s_0,\ldots,s_n/s_0$ among a generating set of its section ratios. For every $q$, the sections
$$
s_0^{q-|\alpha|}\prod_{i=1}^ns_i^{\alpha_i},\qquad \alpha\in\mathbb N^n,\quad |\alpha|\leq q,
$$
are linearly independent by <algebraic independence>. Therefore $h^0(X,qjD)\geq\binom{q+n}{n}$, giving condition (1) of part (iii). This is the <birational linear system criterion for bigness>.
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