Ample divisors on the blow up of 𝐏ⁿ at points

Abstract

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