PomeProposition 12.27.7.9124282E25D3 Let $S$ be obtained from $\mathbb P^2$ by nine successive point blowups, each centered on the strict transform of a fixed smooth cubic. The final strict transform $D$ of the cubic is a smooth elliptic anticanonical curve with $D^2=0$, and the anticanonical bundle has a smooth semipositive metric whose curvature has rank at most one, as in proposition 12.27.5. The anticanonical bundle is semiample exactly when $N=\mathcal O_ D(D)$ is torsion. If $N$ has exact order $m$, then $|mD|$ is basepoint-free and $|kD|=\{ kD\} $ for $1\leq k<m$. If $N$ is not torsion, then $|kD|=\{ kD\} $ for every $k\geq 1$.
Proposition 12.27.79124282E25D3
Proof. A blowup at a point of the cubic replaces the class of the cubic by its pullback minus the exceptional curve, exactly as it does for the anticanonical class. The strict transform remains isomorphic to the cubic, and its self-intersection decreases by one. After nine steps, $D$ is anticanonical and $D^2=9-9=0$. Theorem 12.27.3, Proposition 12.27.5 give the metric.
For $k\geq 1$ we use the restriction sequences
\[ 0\longrightarrow \mathcal O_ S((k-1)D) \longrightarrow \mathcal O_ S(kD) \longrightarrow N^ k\longrightarrow 0. \]
A nontrivial line bundle of degree zero on an elliptic curve has no nonzero section, since such a section would have a zero divisor of degree zero and would trivialize the bundle. If $N$ is not torsion, induction on $k$ gives $h^0(S,\mathcal O_ S(kD))=h^0(S,\mathcal O_ S)=1$. This proves that every $|kD|$ is the single divisor $kD$, and in particular that the anticanonical bundle is not semiample.
Suppose now that $N$ has exact order $m$. For $1\leq k<m$ we have $H^0(D,N^ k)=0$, and $H^1(D,N^ k)=0$ by Serre duality on the elliptic curve. Since $S$ is rational, $H^1(S,\mathcal O_ S)=0$. The same sequences give $h^0(S,\mathcal O_ S(kD))=1$ and $H^1(S,\mathcal O_ S(kD))=0$ for $0\leq k<m$. For $k=m$ the restriction map to $H^0(D,N^ m)$ is therefore surjective, and the constant section of $N^ m\simeq \mathcal O_ D$ lifts to a section of $\mathcal O_ S(mD)$ that does not vanish on $D$. Together with $s_ D^ m$, this section generates $\mathcal O_ S(mD)$ everywhere. Thus $|mD|$ is basepoint-free, while no smaller positive multiple of $D$ moves.
Finally, non-torsion configurations exist, with nine distinct centers. Fix the first eight points on the cubic and vary the ninth. Under the identification of $D$ with the cubic, the normal bundle is $\mathcal O_ D(3)\otimes \mathcal O_ D(-p_1-\cdots -p_9)$, where $\mathcal O_ D(3)$ is the restriction of $\mathcal O_{\mathbb P^2}(3)$. As $p_9$ varies, these bundles run through the whole group of line bundles of degree zero, by the Abel map of the elliptic curve. The torsion subgroup is countable, whereas the curve is uncountable, so even after excluding the first eight points there are non-torsion choices. For them, corollary 12.27.6 gives nonclosed backgrounds of order one, although the anticanonical bundle is not semiample. $\square$
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