General blowings-up of P2 and injectivity of Gauss maps

We consider the blowing-up Y-k of the projective plane along k general points P-1,...,P-k. Let pi (k): Y-k--> P-2 be the projection map and E-i the exceptional divisor corresponding to P-i for 1 less than or equal toi less than or equal tok. For m greater than or equal to2 and k less than or equal tom(m+3)/2-4 let M-k be the invertible sheaf pik*(O-P2(m)) x O-Yk(-E1-...-E-k) on Y-k, and let phi (k): Y-k--> P-N be the morphism corresponding to M-k. As phi (k) is a local embedding, the Gauss map gamma (k) corresponding to M-k is defined on Y-k by gamma (k)(x)=(d(x)phi (k))(T-x(Y-k)) for all x is an element ofY(k). We prove that this Gauss map gamma (k) is injective.