Towards a Halphen theory of linear series on curves

Abstract

A linear series g δ N g^{N}_{\delta } on a curve C ⊂ P 3 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 δ \delta , in terms of deg( C C ), g( C C ) and of the number s = min { i : h 0 I C ( i ) ≠ 0 } 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.