Completeness and non-speciality of linear series on space curves and deficiency of the corresponding linear series on normalization

Abstract

Let Г be a curve ofP r (r⩾3) of degree d, C its normalization and $$\mathcal{A}\mathop \supset \limits_{ - /} $$ , I(Γ) a saturated, homogeneous ideal of k[X0, ...,X r]. In this paper we show that, if N ⩾ 0 is an integer such that, for n⩾N, the linear series cut out on Γ by the hypersurfaces of degree n is complete and non-special, then the deficiency of the linear series cut out on C by the hypersurfaces ofA n,forn>N, is independent ofn and can be explicitly calculated;this is the case, for instance, whenN=d−r+1, and when N=∑ni −r−1 (under suitable conditions) if Γ is a component of the complete intersection of r−1 hypersurfaces of degrees ni.