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 $n = 4$)?
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.
