It is well-known that $H^2(dP_n, Z)$ for a del Pezzo surface of degree $9-n$ which is $\mathbb{P}^2$ blown up at $n$ generic points, is encoded by the exceptional Lie algebra $E_n$. However, the Mori cone of effective curves ($-1$-curves here) on $dP_{9-n}$ is also related to $E_{n}$ as follows.
The number of generators of the Mori cone for $dP_{n}$ is for $n=1,2,\ldots,8$, respectively $2,3,6,10,16,27,56,240$. These should be related to the dimension of the irreps of $E_n$.
For example, the 27 (which is also the famous 27 lines on the cubic surface) is the dimension of the fundamental representation of $E_6$. The 16 is the dimension of the adjoint of $SO(10) = E_5$, the 10, that for $SU(5) = E_4$, etc. (btw, why are some fundamental and some are adjoint representations)?
However the highest two require some work: 56 is twice the dim of fundamental of $E_7$ and and $240 = 248 - 8$ where 248 is the dimension of the fundamental and 8 is the rank of $E_8$
Could someone explain why the last two $n=7,8$ require this "fudge"?
