= Castelnuovo contraction criterion
{c}
{title2=$C\cong\mathbb P^1,\quad C^2=-1$}
A <smooth rational curve> with self-intersection $-1$ on a <smooth projective surface> over an algebraically closed field contracts to a smooth point of another <smooth projective surface>. The contraction is a <birational morphism> and an isomorphism off the curve, with inverse a <blowup of a smooth algebraic surface>. This algebraic criterion is valid in arbitrary characteristic and should not be confused with Castelnuovo's rationality criterion.
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