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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.

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.

A smooth rational curve with self-intersection $-1$ on a smooth projective surface over an algebraically closed field contracts to a smooth point of another smooth projective surface. The contraction is a birational morphism and an isomorphism off the curve, with inverse a blowup of a smooth algebraic surface. This algebraic criterion is valid in arbitrary characteristic and should not be confused with Castelnuovo's rationality criterion.

Put $L={\mathcal{O}}_X(D)$ and let $s_D$ be the canonical global section cutting out the effective Cartier divisor $D$. The divisor restriction exact sequence givesThe hypothesis says $L{|}_D$ is an ample line bundle. By Serre vanishing, $H^1(D,L^m{|}_D)=0$ for all sufficiently large $m$, so the long exact sequence in sheaf cohomology makessurjective for all such $m$. These are finite-dimensional vector spaces. Their dimensions form a nonincreasing sequence of nonnegative integers, so the maps are isomorphisms from some point on. Exactness then implies that the restrictionis surjective for all sufficiently large $m$.

Choose such an $m$ for which $L^m{|}_D$ is also a globally generated line bundle. Lift a generating collection of its global sections to $X$. At each point of $D$, one lift has nonzero image in the one-dimensional residue-field fibre, hence generates the stalk of $L^m$ by Nakayama lemma. Outside $D$, $s_D^m$ is nowhere zero and generates $L^m$. Together these sections generate it everywhere. ThereforeThis proof works on an arbitrary projective scheme because the defining section of an effective Cartier divisor is a non-zero-divisor. If $D$ is empty, $L\cong {\mathcal{O}}_X$ and the conclusion is immediate. Semiampleness is the conclusion: the pullback of a line avoiding the centre of a blowup of a smooth algebraic surface of ${\mathbb{P}}^2$ satisfies the hypothesis on its support but has zero intersection with the exceptional curve, and therefore is not ample.

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.

The Castelnuovo contraction criterion says that a smooth rational curve $C$ on a smooth projective surface with $C^2=-1$ can be contracted by a birational morphism to a smooth point of a smooth projective surface. The contraction is an isomorphism away from $C$, and its inverse is the blowup of a smooth algebraic surface at that point. This is the algebraic contraction criterion, valid over the algebraically closed field here; it is not the rationality criterion bearing Castelnuovo's name.

By (b), every $C\in \mathrm{Neg}(K_X)$ satisfies the hypothesis. Therefore the required morphism exists:The PDF calls $f_1$ a birational map; the conclusion is stronger, since this map is everywhere defined and is a birational morphism.