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A Cartier divisor $D$ on an integral $n$-dimensional projective variety is big when $h^0(X,mD)\ge c{m}^n$ for some $c>0$ and infinitely many positive integers $m$. Equivalently, its Iitaka dimension is $n$. Kodaira's lemma characterizes bigness by an ample-plus-effective decomposition, and the birational linear system criterion for bigness shows why this growth captures the full variety.

Each $M_n$ is a nef divisor. If $M_n^2>0$, the volume of a nef divisor formula gives quadratic growth of $h^0(X,q{M}_n)$. Multiplication by the section of $q{F}_n$ injects this space into $H^0(X,qnH)$, so the birational linear system criterion for bigness makes the Iitaka dimension of $H$ two.

Conversely, if $\kappa (H)=2$, some $|nH|$, and hence its movable part $|M_n|$, has a two-dimensional image. If $M_n^2=0$, the previous part would make $|M_n|$ basepoint-free with image a curve; this is impossible. Since $M_n$ is nef, $M_n^2\ge 0$, hence $M_n^2>0$.

If all $M_n^2=0$ and one $|M_{n^{\prime }}|$ has positive dimension, the preceding part supplies a curve image. The just-proved equivalence excludes dimension two, so $\kappa (H)=1$. Conversely $\kappa (H)=1$ excludes every positive $M_n^2$ and requires a positive-dimensional linear system. Thus

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.