When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space? 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?
 A: CamSar's answer above basically finishes this; I'd just like to add a couple points.


*

*The proof that $-K_X$ is big for $k \leq 8$ isn't really right, because multiplicities at points don't impose independent conditions in general.  For example, two points of multiplicity $2$ do not impose independent conditions on degree $2$ curves (the dimension count says there shouldn't be any, but there is a double line).  Saying when the conditions are independent is very hard, and is the subject of the SHGH conjecture.  A correct proof is easier anyway: for $k \leq 8$, $-K_X$ moves in a pencil, so is nef.  But $(-K_X)^2 = 9-k > 0$, so $-K_X$ is big.  (Of course it is in fact ample -- these are del Pezzo surfaces!  This is spelled out in Hartshorne, I think.)

*It's worth stressing the punchline: if $-K_X$ is ample (even big and nef), then $X$ is a MDS.  This is proved in all dimensions in [BCHM]; the surface case is probably much easier.

*If $k \geq 9$, it is not a Mori Dream Space, because there are infinitely many $(-1)$-curves and so the effective cone is not rational polyhedral.  $-K_X$ not big alone doesn't prove anything; for example, if you blow up any number of points on a line in $\mathbb P^2$ the result is MDS but the anticanonical is not big.
A: 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.
