Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let $\pi:X\to\mathbb{P}$ be the composition of blowing up in $L_1$, then blowing up in the strict transform of $L_2$, and so on.

Let $E_i\:=\pi^{-1}(L_i)$ and $E=E_1+\cdots+E_k$ the exceptional divisor. Now, any divisor on $X$ is of the form

$H=\pi^\ast(D) + \sum_{i=1}^k a_i E_i$

I am wondering when $H$ is ample - I am very much willing to assume that $D$ is ample, and I am looking for a condition that depends mostly on the $a_i$. If this is still too general, I would like to know if an anticanonical divisor

$-K_X=-K_{\mathbb{P}} - \sum_{i=1}^k (r_i-1) E_i$

on $X$ is ample.

The above is the least general scenario that I am willing to study - more generally, what are the "best" sufficient conditions for ampleness of a divisor on a nonsingular blow-up?