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By Kodaira's lemma for the big divisor $K_X$, choose an integer $q>0$, an ample Cartier divisor $A$, and an effective divisor $E$ withIf an integral curve $C$ is not a component of $E$, nonnegative intersections of distinct curves giveThus every curve with $K_X\cdot C<0$ is one of the finitely many components of $E$. In particular,This is the surface case of negative curves of a big real divisor lie in finitely many divisors. The proof does not assert that every negative curve on $X$ belongs to this finite set; it concerns curves negative against this particular big canonical class.

Fix any ample divisor $A$. Choose $q\gg 0$ so that both $qA$ and $(q-1)A$ have nonzero global sections, and choose $F\in |qA|$. Condition (1) and the section subtraction lemma for big divisors yield an integer $j>0$ with $jD-qA\sim E_0\ge 0$. Adding an effective member of $|(q-1)A|$ gives $jD\sim A+E$. Thus (1) implies the stronger condition (2').

If $jD\sim A+E$ with $A$ ample and $E$ effective, multiplication by the section of $qE$ injects $H^0(X,qA)$ into $H^0(X,qjD)$. The positive leading coefficient of the ample Hilbert polynomial supplies condition (1) along the infinite sequence $m=qj$. Condition (2) implies (3). Conversely, if $jD\equiv A+E$, put $P=jD-E$. The Cartier divisor $P$ is numerically equivalent to $A$, hence ample by Kleiman's criterion; the actual equality $jD=P+E$ gives (2).

We have proved (1)$\Rightarrow$(2')$\Rightarrow$(2)$\Rightarrow$(1), and (3)$\Rightarrow$(2)$\Rightarrow$(3). Also (2') implies (3'), while (3') implies (3). Hence all five conditions are equivalent. They characterize a big divisor; the ample-plus-effective formulation is Kodaira's lemma. The inconsistent use of $n$ and $j$ in the printed multiplier is resolved by using $j$ throughout.

The positivity of a Cartier divisor can be measured by sections, intersection numbers, or its numerical class. Ample divisors give projective embeddings after taking a multiple; nef divisors are their numerical limits; big divisors have the maximum possible order of section growth. Nefness and bigness are different conditions.

If $D$ is a big divisor and $F$ an effective Cartier divisor, infinitely many $m$ have $H^0(X,mD-F)\ne 0$. The divisor restriction exact sequence bounds the dimension lost upon restriction to $F$ by $O(m^{n-1})$, using the polynomial bound for sections of a fixed divisor; this cannot exhaust the $c{m}^n$ sections along the infinite growth sequence.