Fix a hyperplane H ;⊂ ..., n > 3, and a finite set S ;⊂ H. We give conditions on the integers d, g and #(S) such that there exists a smooth and connected curve X ;⊂ ... with deg(X) = d, pa(X) = g and S ;⊂ X ∩H. When d = #(S) we may take g up to order 2d/n, d » 0, when S is in linear general position. We also prove the existence of X with h1(NX (-1)) = 0 if n ≥ 8, g is odd and 2d ≥ (n-3)g+n+11.