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?