Ample divisors on the blow up of $\mathbf {P}^n$ at points

Abstract:

Fix integers $n,k,d$ with $n\ge 2,d\ge 2$ and $k>0$; if $n=2$ assume $d\ge 3$. Let $P_1,\dotsc ,P_k$ be general points of the complex projective space $\mathbf {P}^n$ and let $\pi :X\to \mathbf {P}^n$ be the blow up of $\mathbf {P}^n$ at $P_1,\dotsc ,P_k$ with exceptional divisors $E_i:=\pi ^{-1}(P_i)$, $1\le i\le k$. Set $H:=\pi ^*(\mathbf {O}_{\mathbf {P}^n}(1))$. Here we prove that the divisor $L:=dH-\sum _{1\le i\le k}E_i$ is ample if and only if $L^n>0$, i.e. if and only if $d^n>k$.