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For an integral projective curve on a smooth projective surface, blowing up a point of multiplicity $m$ gives $C^{\prime }={\pi }^{\ast }C-mE$. The intersection formula for blowing up a surface and canonical divisor formula for a surface blowup yield $C^{\prime 2}=C^2-m^2$ and $K_{X^{\prime }}\cdot C^{\prime }=K_X\cdot C+m$. Substitute both in the arithmetic adjunction formula on a smooth surface. This does not require the transformed curve to be smooth.

Blow up the distinct singular points of the integral projective curve, once at each original point, and let $C^{\prime }$ be its strict transform of an algebraic subvariety. Write $H$ for the pullback of a projective line and $E_i$ for the algebraic exceptional divisors. The multiplicity of a plane curve at a point and the intersection formula for blowing up a surface giveThe canonical divisor formula for a surface blowup gives $K=-3H+{\sum }_i{E}_i$. Applying the arithmetic adjunction formula on a smooth surface, rather than a formula requiring a smooth strict transform, yieldsand henceThe strict transform of an algebraic subvariety is still an integral projective curve. Its arithmetic genus is $h^1(C^{\prime },{\mathcal{O}}_{C^{\prime }})\ge 0$, even if some singularities remain. Therefore the plane curve singularity multiplicity bound isThe argument uses only one blowup of a smooth algebraic surface at each original singular point; it does not assume these blowups resolve every singularity.

Part (i) first shows that $D-C$ is effective. Also $C^2<0$: writing $D=aC+E$ with $a\ge 1$ and $E$ effective without $C$ as a component gives $D\cdot C=a{C}^2+E\cdot C$, where $E\cdot C\ge 0$.

For $0\le j<m$,Every effective member of $|mD|$ therefore contains $C$; after subtracting it, the same reasoning applies again, until $m$ copies have been subtracted. Conversely adding $mC$ to an effective member of $|m(D-C)|$ is allowed. HenceFor $m=0$ this is just the identity system $|0|$. The extra intersection hypothesis is needed: on the Hirzebruch surface ${\mathbb{F}}_2$, with negative section $S$ and fibre $F$, take $D=S+F$. Then $D\cdot S=-1$ but $(D-S)\cdot S=1$. Here $h^0(2D)=4$ while $h^0(2F)=3$, so removing $2S$ loses a section.

For the point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$ with exceptional curve $E$, the canonical divisor formula for a surface blowup and the intersection formula for blowing up a surface giveOne must not assume $K_{\tilde{X}}$ itself is effective. Instead, if an effective member of $|m{K}_{\tilde{X}}|$ exists, then for each $0\le j<m$ its class after removing $jE$ has intersection $-m+j<0$ with $E$. The same fixed-component argument removes exactly the required $mE$, givingas an isomorphism, also when both sides are zero. Since $X$ is normal and $\psi$ is proper and birational, ${\psi }_{\ast }{\mathcal{O}}_{\tilde{X}}={\mathcal{O}}_X$. The projection formula for sheaves therefore identifies the space on the left with $H^0(X,{\mathcal{O}}_X(m{K}_X))$. ConsequentlyThe PDF writes equality of spaces; this is the canonical identification just described, rather than literal equality of spaces on different schemes. This proves blowup invariance of plurigenera in arbitrary characteristic.