TITLE: Why only some del Pezzo are toric?
QUESTION [7 upvotes]: Let us define smooth del Pezzo surfaces $dP_r$ as the blowup of $r$ generic points in $\mathbb{CP}_2$. One can show that if we request $dP_r$ to be Fano, then $r=0,...,8$. In theoretical physics literature is common to define also $dP_9$, which is the blowup of $9$ points in $\mathbb{CP}_2$ and refer to this as well as a del Pezzo surface, while as far as I understand mathematicians would call this a generic rational elliptically fibered surface.
Now, the question is why only $dP_r$ for $r=0,1,2,3$ are toric varieties? This statement is given for example in the appendix A.3 of http://inspirehep.net/record/1707635
For example, consider the fan of $\mathbb{CP}_2$. In order to perform the $i$-th blow-up, at the level of the fan, we add a new 1-dimensional cone which is representing the exceptional divisor $E_i$, associated to the new $\mathbb{CP}_1$. Which is the obstruction from keeping adding cones also for $r>3$? Why does the toric description fail?

REPLY [3 votes]: You have no problem for adding cones from a toric surface $\mathbb{P}^2$ or any surface obtained by blowing-up points on it. 
However, there are only three toric points on $\mathbb{P}^2$, so once you have blown-up these $3$ points, the toric points that you see on your surface lie on $(-1)$-curves, and thus do not correspond anymore to points of $\mathbb{P}^2$. You can still blow-up them, and it corresponds to what you expect on the fan point of view, but you no longer get a del Pezzo surface, as there are $(-2)$-curves.
In general, if you blow-up more than $8$ points of $\mathbb{P}^2$ in (very) general points, you get a non-toric surface with only curve of self-intersection $\ge -1$ and such that negative curves should be $(-1)$-curves (it is in fact a conjecture, see https://arxiv.org/pdf/math/0512631.pdf ), but this is no longer del Pezzo as the anti-canonical divisor is no longer ample, only nef and not big.