Del Pezzo surfaces are obtained by blowing up $1 \leq k \leq 8$ points on general position in $\mathbb{P}^2$. What does it happen when the number of points is larger than nine? In this sense, Beauville's book in surfaces presents the topic in the context of linear system of cubics: Nine points in the plane determine a cubic curve, and del Pezzo surfaces $S_{9-k}$, with $k\leq 6$, are embedded into $\mathbb{P}^{9-k}$ by the linear system of cubics through the $k $ points. Is there a nice interpretation of the surfaces obtained by linear system of plane curves of degree $d$?
I suppose this is well known but I cannot find a reference. Thanks!!
Best Answer
The canonical divisor of the blow up $\pi: X\to \mathbb P^2$ at $k$ ordinary points is $$ K_X = -3\pi^*L +\sum_{i=1}^k E_i, $$ where $L\subset \mathbb P^2$ is a hyperplane and $E_i$ is an exceptional curve of the first kind. Choosing the representatives right and an easy computation shows that $$ K_X^2 = 9 - k, $$ so $-K_X$ could possibly be ample only if $0\leq k\leq 8$.
As Artie points out, embedding into $\mathbb P^{9-k}$ only works for $0\leq k\leq 6$. It is easy to see that this cannot be true for $k>6$ as then $9-k\leq 2$ and there is no way you can embed a surface different from $\mathbb P^2$ into $\mathbb P^2$, $\mathbb P^1$, or $\mathbb P^0$.
So, the interesting question is what you get if $k=7$ or $8$.
It is relatively easy to see that $-K_X$ is not very ample (unlike in the $0\leq k\leq 6$ case): By looking at the short exact sequences, $$ 0\to \pi^*\omega_{\mathbb P^2}^{-1}(-\sum_{i=1}^r E_i) \to \pi^*\omega_{\mathbb P^2}^{-1}(-\sum_{i=1}^{r-1} E_i)\to \mathscr O_{E_r}\to 0 $$ one can see easily that $$ \dim H^0(X, \omega_X^{-1}) = 10-k. $$ In fact the $\mathbb P^{9-k}$ above is just the projectivization of this linear space.