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?
Best Answer
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 (Proceedings of the Amer. Math. Soc. 130, 2002), proved the following (non optimal) result:
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 , Manuscripta Mathematica 93 (1997).