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?



Ample divisors on blown-up projective space
Your question is actually far too general, so let me assume $n=2$. Also in this case, there are only partial results.


In the case where all $a\_i$ are equal to $1$, Kurchle and (independently) Xu showed that 
$$H=\pi^\*(dL) - \sum\_{i=1}^r E\_i$$
 (where $L$ is the class of a line) is ample, provided that $H^2 > 0$ and $d \geq 3$.


Later, Szemberg and Tutaj-Gasinska, in their paper [General blow-ups of the projective plane](http://www.ams.org/journals/proc/2002-130-09/S0002-9939-02-06488-2/S0002-9939-02-06488-2.pdf) (Proceedings of the Amer. Math. Soc. 130, 2002), proved the following (non optimal) result:



> 
> **Theorem.** Let $X$ the blow-up of $\mathbb{P}^2$ at $r$ *general* points and let $k\geq 2$ and $r$ be integers such that $d \geq 3k+1$. If $r \leq \frac{d^2}{k^2} -1$, then the line bundle $$H=\pi^\*(dL) -k \sum\_{i=1}^r E\_i$$ is ample. 
> 
> 
> 


I refer you to Szemberg-Tutaj-Gasinska paper for more details on this problem and on its relations with the so-called Nagata Conjecture.


In higher dimensions, there is a paper by Angelini that generalizes the result of Xu for $n=3$, in the case where all blown-up subvarieties are points. See [Ample divisors on the blow up of $\mathbb{P}^3$ at points](http://www.springerlink.com/content/n712516h42571181/) , Manuscripta Mathematica 93 (1997).


