Non-special projectively normal line bundles on general k-gonal curves

Abstract

Abstract. Let X be a sufficiently general smooth k-gonal curve of genus g and R ∈ Pic(X) the degree k spanned line bundle. We find an optimal integer z > 0 such that the line bundle R ⊗z is very ample and projectively normal.1. Introduction. Let X be a smooth and connected projective curve of genus g, and L a spanned line bundle on X. Let h L : X → P(H 0 (X, L) * ) denote the morphism induced by the complete linear system |L|. Here we assume that X is a sufficiently general k-gonal curve and explore the projective normality of non-special very ample line bundles on X.Our main result is the following theorem giving the projective normality of several non-special line bundles L on general k-gonal curves of genus g. Indeed, often deg(L) 3g/2 and hence this is outside the range covered in [4]. The result (and its proof) is very different from the case of special line bundles considered in [6] (see in particular lines 4-8 of page 189 of [6]) and the references therein.