Canonical divisor formula for a surface blowup
= Canonical divisor formula for a surface blowup
{title2=$K_{\widetilde X}=\psi^*K_X+E$}
For a point <blowup of a smooth algebraic surface> $\psi:\widetilde X\to X$ over an algebraically closed <field>, with exceptional curve $E$, $K_{\widetilde X}=\psi^*K_X+E$ after choosing compatible representatives. In local coordinates $y=xu$, the differential $dx\wedge dy=x\,dx\wedge du$ has one additional zero along $E$. Thus the formula is valid in every characteristic.