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.

For a point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$ over an algebraically closed field, with exceptional curve $E$, $K_{\tilde{X}}={\psi }^{\ast }{K}_X+E$ after choosing compatible representatives. In local coordinates $y=xu$, the differential $dx\wedge dy=xdx\wedge du$ has one additional zero along $E$. Thus the formula is valid in every characteristic.