Very Strange Projective Curves

Let Y ? Pⁿ, n ≥ 3, be an integral non-degenerate very strange projective curve, i.e. assume that the general hyperplane section of Y is not in linearly general position; hence we are in characteristic p. Let π: X → Y be the normalization. Set d? deg(Y), g? Pa(X) and L? π* (Oy(1)). Here we prove that d > 2g?2 and in particular h¹(X,L) = 0 and h⁰(X,L) = d+1?g ≥ (d?1)/2 > n+1.