Castelnuovo's contraction theorem

Remarks

The idea of the proof is to choose a very ample divisor $H$ on $X$ with $H^1(X, \mathcal{O}_X(H)) = 0$, put $m = H \cdot E$, and use the linear system $\abs{H + mE}$. It has degree $0$ on $E$, so it contracts $E$ to a point, and a computation of sections shows that the image surface is smooth at that point. Example: a smooth cubic surface over $k$ contains $27$ lines, each a $(-1)$-curve, and contracting six disjoint lines among them gives $\mathbb{P}^2$. The condition $E \cdot E = -1$ cannot be weakened to $E \cdot E < 0$. A smooth rational curve with $E \cdot E = -2$ can sometimes be contracted, but the image point is then singular, an $A_1$ singularity.

Going deeper

For a student

The theorem is the basic step of the minimal model program for surfaces. Contracting $(-1)$-curves one at a time lowers the Picard number, so the process stops, at a surface with no $(-1)$-curves. So every smooth projective surface is obtained from a surface without $(-1)$-curves by finitely many blow-ups of points. The surfaces without $(-1)$-curves are then sorted by the Enriques–Kodaira classification.