Let $p\_1,...,p\_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X\_k = Bl\_{p\_1,...,p\_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X\_k$ is toric and hence a Mori Dream Space. The question is:


for which $k$ is $X\_k$ a Mori Dream Space?



When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space?
If $S$ is a smooth, projective, rational surface and $-K\_S$ is big, then $S$ is a Mori Dream Space. I will prove the following: if $k\leq 8$ then $X\_k$ is a Mori Dream Space.


The anticanonical of $X\_k$ is 
$$-K\_{X\_k} = 3H-E\_1-...-E\_k.$$
Therefore $-mK\_{X\_k} = 3mH-mE\_1-...-mE\_k$. Therefore the anticanonical correspondes to the linear system of plane curves of degree $3m$ with points of multiplicity $m$ at $p\_1,...,p\_k$. Now, $h^0(\mathbb{P}^2,\mathcal{O}(3m)) = \binom{3m+2}{2}$. Furthermore a point of multiplicity $m$ imposes $\binom{m+1}{2}$ conditions. Considering the difference $\binom{3m+2}{2}-k\binom{m+1}{2}$ we get:
$$\lim\_{m \to +\infty}\frac{h^0(X\_k,-mK\_{X\_k})}{m^2} = \lim\_{m \to +\infty}\frac{(9-k)m^2+(9-k)m+2}{2m^2} = \frac{9-k}{2}.$$
Finally $9-k>0$ if and only if $k<9$. If $k = 9$ then $-K\_{X\_9}$ is nef but $K\_{X\_9}^2 = 0$. Then $-K\_{X\_9}$ is not big.


