Effective divisors on the blow-up of $\mathbb{P}^n$ Let $X^n_r$ the blow-up of $\mathbb{P}^n$ at $r$ points in 
very general position.
(1) It is known in general the Effective Cone or the 
Numerical Effective Cone of those algebraic variety?
Let $X$ be an algebraic variety.
(2) Where i can find equivalent conditions for a 
divisor class $D\in Pic (X)$ to be effective?
Thanks in advance !
 A: As explained in the comment of Ruadhai Dervan, the case where $X$ is not Fano (or even worse, when $X$ is not weak-Fano) is probably hard to deal with. 
The Picard group of $X_r^n$ is generated by $H$, the pull-back of a hyperplane, and by $E_1,\dots,E_r$, the divisors associated to the points blown-up.
In dimension $2$, Nagata's conjecture corresponds to say that if $r> 9$ and $dH-\sum_{i=1}^r  a_iE_i$  is effective, then $\sum a_i< \frac{d}{\sqrt{r}}$ (the result is false for $r\le 8$, which corresponds to the Fano case and for $r=9$, as you take a cubic through the points). This is true when $r$ is a square, and proved by Nagata, but still open if $r$ is not a square and $r>9$. There are however partial results in the direction of the conjecture, that you can find on the web by looking for "Nagata conjecture for curves".
In dimension higher, I don't know what are the results when the variety is not Fano.
You can also describe when $X_r^n$ is Fano, or only weak-Fano, depending on $r$ and $n$. 
In "Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links." Proc. Lond. Math. Soc. (2012) 105(5): 1047-1075, S. Lamy and myself considered the case of blow-ups of curves in $\mathbb{P}^3$, but also did the easier case of points (see section 2.7), which is basically an exercise. 
I summarise the result:
For $n\ge 3$, $X_r^n$ is Fano if and only if $r=1$ and is weak-Fano (and not Fano) if and only if $n=3$ and $2\le r\le 7$.
You can also have a look at the texte and see the conditions that you need to put on the points so that you obtain a weak-Fano $3$-fold.
A: The nef cone of $X_r^n$ is particularly well behaved when $X_r^n$ is a Mori dream space. In this case it has a nice decomposition made of the nef cones coming from its small $\mathbb{Q}$-factorial modifications. It is known that.


*

*$X_{r}^n$ for $n\geq 5$ is a Mori dream space if and only if $r\leq n+3$,

*This bound can be improved for $n=3,4$. Indeed $X_r^3$ is a Mori dream space if and only if $r\leq 7$, and $X_r^4$ is a Mori dream space if and only if $r\leq 8$,

*finally $X_r^2$ is a Mori dream space if and only if $r\leq 8$ (these are Del Pezzo surfaces).


You can find this here:
http://arxiv.org/abs/math/0505337
