Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 134 / 2

On a smooth algebraic surface, distinct integral curves have nonnegative intersection numbers, because their intersection multiplicities are nonnegative. For any member $E\in |D|$ of the complete linear system of a divisor, $E\cdot C=D\cdot C<0$. Hence $C$ must be a component of $E$; otherwise the intersection would be nonnegative.

Subtracting one copy of $C$ from every effective member gives precisely the effective members linearly equivalent to $D-C$. Conversely adding $C$ gives a member of $|D|$. Thus $C$ is a fixed component andEquivalently, multiplication by its defining section induces an isomorphism $H^0(X,{\mathcal{O}}_X(D-C))\stackrel{\sim }{\to }{H}^0(X,{\mathcal{O}}_X(D))$. The original PDF has this linear-system equality; the TeX loses the bars and reduces it to an uninformative divisor identity.

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.

Because $C\cong {\mathbb{P}}^1$ and $\mathrm{deg}{\mathcal{O}}_C(C)=C^2=0$, its normal line bundle is ${\mathcal{O}}_C(C)\cong {\mathcal{O}}_{{\mathbb{P}}^1}$. Thus every ${\mathcal{O}}_C(mC)$ is trivial and has $H^1=0$. The divisor restriction exact sequence gives, for every $m\ge 1$,The maps $H^1(X,(m-1)C)\to H^1(X,mC)$ are surjective. Their nonnegative finite dimensions eventually stabilize, so for every sufficiently large $m$ restriction $H^0(X,mC)\to H^0(C,{\mathcal{O}}_C)=k$ is surjective. A lift of $1$ is nowhere zero along $C$, and the canonical section of $mC$ is nowhere zero outside $C$. Together they generate ${\mathcal{O}}_X(mC)$. Therefore $C$ is semiample.

For those same large $m$, the exact sequence of global sections givesThus $h^0(X,mC)=m+O(1)$, and the Iitaka dimension is exactly one:Equivalently, its basepoint-free multiple defines a morphism to a curve: it is nonconstant because its sections grow, and cannot have two-dimensional image because $(mC{)}^2=0$. This is the semiampleness of a square-zero rational curve; it uses no characteristic-zero vanishing theorem.

By Kodaira's lemma for the big divisor $K_X$, choose an integer $q>0$, an ample Cartier divisor $A$, and an effective divisor $E$ withIf an integral curve $C$ is not a component of $E$, nonnegative intersections of distinct curves giveThus every curve with $K_X\cdot C<0$ is one of the finitely many components of $E$. In particular,This is the surface case of negative curves of a big real divisor lie in finitely many divisors. The proof does not assert that every negative curve on $X$ belongs to this finite set; it concerns curves negative against this particular big canonical class.

Let $C\in \mathrm{Neg}(K_X)$. The arithmetic adjunction formula on a smooth surface givesWe first exclude $C^2\ge 0$, following the PDF's hint and the finiteness proved in (a).

If $C^2=0$, adjunction and $K_X\cdot C<0$ force $p_a(C)=0$ and $K_X\cdot C=-2$. An integral projective curve of arithmetic genus zero is a smooth rational curve: normalization and the nonnegative singularity-length correction show both normalization genus and singularity correction vanish. Part (iii) makes $C$ semiample. Choose a basepoint-free $|mC|$. Over the infinite algebraically closed field, a general section avoids containing any of the finitely many curves of $\mathrm{Neg}(K_X)$ as a component. Its effective divisor $G$ then has $K_X\cdot G\ge 0$, but $K_X\cdot G=m{K}_X\cdot C<0$, a contradiction.

If $C^2>0$, Riemann–Roch theorem for algebraic surfaces and Serre duality show $h^0(X,mC)$ is unbounded. Indeed $h^2(X,mC)=h^0(X,K_X-mC)=0$ for $m\gg 0$, because the latter divisor has negative intersection with a fixed ample divisor. HenceThere is therefore some $m$ with $h^0(X,mC)>h^0(X,(m-1)C)$, giving a section whose divisor does not contain $C$. The canonical section of $mC$ does not vanish identically along any other curve. A general linear combination of these two sections consequently contains none of the finite set $\mathrm{Neg}(K_X)$, again contradicting its negative canonical intersection. Thus $C^2<0$.

Now put $a=-C^2\ge 1$ and $b=-K_X\cdot C\ge 1$. Adjunction givesIt follows that $a=b=1$ and $p_a(C)=0$. HenceThe general-section argument only avoids finitely many proper linear subspaces, so it works over any algebraically closed field, including positive 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.