Very ample linear systems on blowings-up at general points of projective spaces

Let P-n be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t greater than or equal to 3 consider the invertible sheaf O(t) on P-n (Serre twist of the structure sheaf). Let N = ((t+n)(n))the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 less than or equal to k less than or equal to N - (2n + 2). Let P-1,... P-k be general points on P-n and let pi: X --> P-n be the blowing-up of P-n at those points. Let E-i = pi(-1)(P-i) with 1 < i less than or equal to k be the exceptional divisor. Then M = pi* (O(t)) circle times O-x(-E-1 - ... - E-k) is a very ample invertible sheaf on X.