Towards a Halphen theory of linear series on curves

A linear series g(delta)(N) on a curve C subset of P-3 is primary when it d oes not contain the series cut by planes. For such series, we provide a low er bound for the degree delta, in terms of deg(C), g(C) and of the number s = min {i : h(0)I(C) (i) not equal 0 }. Examples show that the bound is sha rp. Extensions to the case of general linear series and to the case of curv es in higher projective spaces are considered.