Plurigenus ($P_m(X)=h^0(X,m{K}_X)$)

For a smooth projective variety, its $m$th plurigenus is $P_m(X)=h^0(X,{\omega }_X^{\otimes m})$, for $m\ge 1$. These dimensions describe pluricanonical linear systems. Blowup invariance of plurigenera proves their invariance under a point blowup on a smooth surface.

For a point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$, where $X$ is a smooth projective surface over an algebraically closed field, multiplication by the exceptional section to power $m$ identifies $H^0(\tilde{X},m{\psi }^{\ast }{K}_X)$ with $H^0(\tilde{X},m{K}_{\tilde{X}})$. 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 ${\psi }_{\ast }\mathcal{O}=\mathcal{O}$ identify the remaining sections with those on $X$. No effectivity of $K_X$ itself is assumed.