The <Castelnuovo contraction criterion> says that a <smooth rational curve> $C$ on a <smooth projective surface> with $C^2=-1$ can be contracted by a <birational morphism> to a smooth point of a <smooth projective surface>. The contraction is an isomorphism away from $C$, and its inverse is the <blowup of a smooth algebraic surface> at that point. This is the algebraic contraction criterion, valid over the algebraically closed field here; it is not the rationality criterion bearing Castelnuovo's name.

By (b), every $C\in\operatorname{Neg}(K_X)$ satisfies the hypothesis. Therefore the required morphism exists:
$$
\boxed{f_1:X\to X_1,\quad f_1(C)=\{p\},\quad X_1\text{ smooth and projective},\quad X\setminus C\cong X_1\setminus\{p\}.}
$$
The PDF calls $f_1$ a birational map; the conclusion is stronger, since this map is everywhere defined and is a <birational morphism>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 134 / 2 / iv / c / Solution

{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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 Incoming links: Castelnuovo contraction criterion