= Blowup invariance of plurigenera
{title2=$P_m(\widetilde X)=P_m(X)$}
For a point <blowup of a smooth algebraic surface> $\psi:\widetilde X\to X$, where $X$ is a <smooth projective surface> over an algebraically closed <field>, multiplication by the exceptional section to power $m$ identifies $H^0(\widetilde X,m\psi^*K_X)$ with $H^0(\widetilde X,mK_{\widetilde X})$. Every effective pluricanonical representative must contain $mE$, since after subtracting $j<m$ copies its intersection with $E$ is $-m+j<0$. The <projection formula for sheaves> and $\psi_*\mathcal O=\mathcal O$ identify the remaining sections with those on $X$. No effectivity of $K_X$ itself is assumed.
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