Ample divisors on the blow up of P-n at points

Fix integers n; k; d with n greater than or equal to 2; d greater than or e qual to 2 and k > 0 ; if n = 2 assume d greater than or equal to 3. Let P,. ..,P-k be general points of the complex projective space P n and let pi : X --> P-n be the blow up of P-n at P-1,...,P-k with exceptional divisors E-i := pi(-1) (P-i), 1 less than or equal to i less than or equal to k. Set H := pi* (O-Pn (1)). Here we prove that the divisor L := dH - Sigma(1 less th an or equal to) (i less than or equal to) k E-i is ample if and only if L n > 0, i.e. if and only if d(n) > k.