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Let $C$ be a smooth cubic curve in $\mathbb P^2$. Consider $9$ general points from $C$. Let $X$ be the blow up of $\mathbb P^2$ at these $9$ points. Then I have the following questions:

  1. Is the strict transform of $C$ the anticanonical divisor of $X$ (Is there a clean way to see this)?

  2. Assuming $(i)$ is true, is it semiample? (The anticanonical divisor can't be ample as the self intersection is then $0$.)

  3. What kind of surface $X$ is? (Is it also a Del Pezzo surface like blowup of $\mathbb P^2$ at fewer than $9$ points?)

Thanks in advance.

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  1. If $\pi:\widetilde{X}\to X$ is the blowup of a smooth surface in a point, then $K_{\widetilde{X}}=\pi^*K_X+E$. On the other hand, if $p$ is a point of multiplicity $r$ on a curve $C$ in a smooth surface $X$, then $\pi^*C = \widetilde{C}+rE$ where again $\pi:\widetilde{X}\to X$ is the blowup of $X$ in $p$. So for your $X$, repeated applications of the first statement give $K_\widetilde{X}=\pi^*(-3H)+\sum_{i=1}^9 E_i$, while repeated applications of the second statement give $\pi^*C=\widetilde{C}+\sum_{i=1}^9 E_i$, then recognizing $C=3H$ and rearranging gives $\widetilde{C}=\pi^*(3H)-\sum_{i=1}^9 E_i=-K_\widetilde{X}$.

  2. Depends! For a very general collection of points, no, it is not semi-ample, but there are configurations of points where this divisor is semi-ample, corresponding to the restriction of $-K$ to $\widetilde{C}$ being linearly equivalent to a sum of torsion points. See here on MO for some good answers and a fun paper to read (if you have the time).

  3. The surface certainly isn't a Del Pezzo surface - the definition of that requires that $-K$ is ample. But you've already mentioned that $K^2=0$, so $-K$ is not ample by Nakai-Moishezon.

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  • $\begingroup$ thank you very much for such a nice answer. Is this surface elliptic? $\endgroup$
    – Sherlock
    Apr 25, 2023 at 12:43
  • $\begingroup$ in both the formulas shouldn't it be canonical bundle of $\tilde X$? $\endgroup$
    – Sherlock
    Apr 25, 2023 at 18:58
  • $\begingroup$ Yes, you're right, that was a small typo. As far as whether it's elliptic, if the points are general, it won't be - the fibers would determine a pencil of cubics on $\Bbb P^2$ which has 9 base points which are on the intersection of two distict cubic curves. But if you choose the points to be general on $C$, then they can't be the intersection points of two distict cubics. $\endgroup$
    – KReiser
    Apr 25, 2023 at 19:40

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