Contracted curve has negative self-intersection ($C^2<0$)

Let a surjective morphism from a smooth projective surface to a projective surface contract an irreducible curve $C$. Pull back an ample divisor $H$ from the target to obtain $L$. The projection formula for algebraic cycles gives $L\cdot C=0$ and $L^2=\mathrm{deg}(f)H^2>0$. The Hodge index theorem for algebraic surfaces makes the intersection form negative definite on the orthogonal complement of this positive-square class. Since an ample divisor on the source meets $C$ positively, its numerical class is nonzero, and $C^2<0$.