= Kodaira's lemma
{c}
A <Cartier divisor> $D$ on an integral projective variety is big exactly when, for every <ample divisor> $A$, some positive integer $j$ satisfies $jD\sim A+E$ with $E$ effective. To prove the forward direction, subtract an effective high multiple of $A$ using the <section subtraction lemma for big divisors>, then add an effective representative of the remaining multiple. For the reverse direction, multiply sections of $qA$ by the section of $qE$ and use the ample <Hilbert polynomial>.
Back to article page
