= Contracted curve has negative self-intersection
{title2=$C^2<0$}
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 $L^2=\deg(f)H^2>0$. 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 $C^2<0$.
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