Blowup of a smooth algebraic surface
= Blowup of a smooth algebraic surface
Over an algebraically closed <field>, the <blowup of a smooth algebraic surface> at a closed point replaces the point by an exceptional <smooth rational curve> $E\cong\mathbb P^1$, with $E^2=-1$. Locally it is the <blowup of the affine plane at the origin>. It is a proper <birational morphism>, an isomorphism off the exceptional curve, and the blown-up surface is again smooth and projective when the original is projective.