Towards a Halphen theory of linear series on curves

A linear series $g^{N}_{\delta }$ on a curve $C\subset \mathbf{P}^{3}$ is primary when it does not contain the series cut by planes. For such series, we provide a lower bound for the degree $\delta$, in terms of deg($C$), g($C$) and of the number $s=\min \{i:h^{0}\mathcal{I}_{C}(i)\neq 0\}$. Examples show that the bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered.