Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let
$$D = aH-b\_1E\_1-...-b\_kE\_k$$
be a divisor on $X$. Are there conditions on $a,b\_1,...,b\_k$ ensuring that $D$ is ample? I just know results of this kind for $n = 2,3$. What about higher dimension? 


In particular is the divisor $D = 3H-E\_1-...-E\_k$ ample if $k\leq n+3$ (perhaps $k\leq 8$ if $n = 4$)? 



Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points
If $k\leq n+3$ and if $n = 3, k = 7$, or $n = 4, k = 8$ then $X$ is a Mori Dream Space. You can find this here <http://arxiv.org/abs/math/0505337>.


In particular this implies that the cone of curves $NE(X)$ is polyhedral and generated by the finitely many classes of curves. Therefore a divisor $D$ in $X$ is ample if and only if $D\cdot C>0$ for any irreducible effective curve in $X$.


Now, let $p\_1,...,p\_k\in\mathbb{P}^n$ be general points, $C\subset\mathbb{P}^{n}$ an irreducible curve of degree $d$, and $m\_i = mult\_{p\_i}(C)$ the multiplicity of $C$ at $p\_i$. If $k\leq n+4$ then 
$$m\_1+...+m\_k \leq 2d.$$ 


If $C = dR\_i$ is a curve in an exceptional divisor $E\_i$ then $D\cdot C = db\_i$, and you get the conditions $b\_i > 0$ for any $i = 1,...,k$.


If $C = dL-m\_1R\_1-...-m\_kR\_k$ is a not contracted then $D\cdot C = ad-b\_1m\_1-...-b\_km\_k$. In the case $b\_1 = ... = b\_k=b$ we can write $D\cdot C = ad-(m\_1+...+m\_k)b \geq ad-2bd$. Therefore we get $a>2b$.


Summing up, if $k\leq n+3$ and if $n = 3, k = 7$, or $n = 4, k = 8$, if $b >0$ and $a>2b$ 
then
$$D = aH-bE\_1-...-bE\_k$$
is ample. In particular in these cases the divisor
$$D = 3H-E\_1-...-E\_k$$
is ample.


Furthermore, if $N = \binom{d+n}{n}$ and $0\leq k\leq N-(2n+2)$, the $dH-E\_1-...-E\_k$ is very ample. You can find this here: <https://cms.math.ca/cmb/v45/coppens8082.pdf>.


In particular, if $k\leq\binom{3+n}{n}-(2n+2)$ then $D = 3H-E\_1-...-E\_k$ is very ample and in particular ample.


