MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
I might be wrong, but if $L_i$ are line and two of those lines have non-empty intersection, consecutive blow-up of those two is not equal to blowing up the union at the same time.(This is was just a general comment and has nothing to do with your question) – Mohammad F. Tehrani Oct 9 '11 at 14:42
Hm, I have to admit I haven't thought about that very thoroughly. I most definitely want to blow up $L_1$, then $\tilde L_2$, and so on. I will edit my post. – Jesko Hüttenhain Oct 9 '11 at 15:19
up vote 7 down vote accepted

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:

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 $$H=\pi^*(dL) + \sum_{i=1}^r kE_i$$ 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 , Manuscripta Mathematica 93 (1997).

share|cite|improve this answer
Thanks a bunch, I also corrected my question. Is there anything known for $n=3$? – Jesko Hüttenhain Oct 9 '11 at 6:01
You are welcome. I added a reference to a result concerning the case $n=3$. – Francesco Polizzi Oct 9 '11 at 8:22

An example in dimension $3$ where $-K_X$ is not ample. Take $X$ the blow-up of $\mathbb{P}^3$ at six general points $p_1,...,p_6$. The anticanonical divisor is $$-K_X = 4H-2E_1-...-2E_6 = 2(2H-E_1-...-E_6).$$ Therefore for any strict transform $\widetilde{L}_{i,j}$ line $L_{i,j} = \left\langle p_i,p_j\right\rangle$ we have $-K_X\cdot\widetilde{L}_{i,j} = 0$. Furthermore, if $\widetilde{C}$ is the strict transform of the twisted cubic $C$ through $p_1,...,p_6$ we have $-K_X\cdot\widetilde{C} = 0$ as well. Therefore $-K_X$ is not ample. However, since the base locus of $|-K_X|$ is zero dimensional $-K_X$ is nef.

On the other hand if we take $p_1,...,p_k$ general points, with $2\leq k\leq 4$ then the $\widetilde{L}_{i,j}$ still have intersection zero with $-K_X$. However, $X$ is toric and in particular log Fano. This means that there exists a reduced simple normal crossing divisor $D$ on $X$ such that $-(K_X+D)$ is ample and $(X,D)$ is klt.

share|cite|improve this answer
Why do you need $6$ points? If $X$ is the blow-up of $k$ points in $\mathbb{P}^n$ with $n\ge 3$ and $k\ge 2$, then $-K_X$ is not ample, by intersecting a line through the points with it. – Jérémy Blanc Mar 20 '14 at 13:24
Of course, I just wanted to provide a case where a curve which is not the strict transform of a line has zero intersection with the anticanonical. – S_Z_S Mar 20 '14 at 13:55
Ah OK. A simple remark: the twisted cubic through $6$ points are equivalent to the lines (sum of three). – Jérémy Blanc Mar 20 '14 at 14:00

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.