Smooth projective surface

A smooth projective surface is a two-dimensional smooth variety admitting a closed embedding in projective space. Its integral curves are Cartier divisors, and their intersections, arithmetic adjunction and point blowups control its birational geometry.

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.