 Solution

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.