= Birational linear system criterion for bigness
A <Cartier divisor> on an integral projective variety is big if and only if some multiple has a <complete linear system of a divisor> giving a rational map birational onto its image. <Kodaira's lemma> embeds a very ample subsystem in a suitable multiple. Conversely, $n$ <algebraically independent elements> among the section ratios give $\binom{q+n}{n}$ independent degree-$q$ section monomials, where $n=\dim X$. Smoothness is not necessary.
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