# On blow up of Projective Plane

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?)

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.
• in both the formulas shouldn't it be canonical bundle of $\tilde X$? Apr 25, 2023 at 18:58
• 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. Apr 25, 2023 at 19:40