Projectively Normal Embedding of ak-Gonal Curve

Normal generation of line bundles on algebraic curves

“…Note that h 1 (C; A) ¿ h 1 (C; L) + 2 and so n ¿ h 1 (C; L) + 1. If n = 1, then h 1 (C; L) = s = 0 and g n 6 1, which contradicts that L is very ample. Therefore d−a=4 and then dim L (R d−a )=1 where L is the morphism associated with L. Hence R d−a fails to impose independent conditions on quadrics in PH 0 (C; L).…”

“…The following lemma will be used to characterize the image curve of C via the map associated with the line bundle KA −1 where A is the line bundle in Lemma 2.2. [6], Lemma 6). The Castelnuovo number has the following property: (d; r) 6 (d − 2; r − 1) for d ¿ 3r − 2; r ¿ 3.…”

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

“…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. Theorem 1.…”

Gonality and Clifford index of projective curves on ruled surfaces

“…In fact, if the covering morphism is simple, then we can apply the Castelnuovo-Severi inequality: it bounds the degree of line bundles which are not composed with the the covering morphism. Thus, in the cases where the covering morphisms are simple, there are some theorems to determine whether a given line bundle is or is not composed with φ ( [2], [11], [15], [10]).…”

“…[11],Lemma 6).For d ≥ 3r − 2, r ≥ 3, we have π(d, r) ≤ π(d − 2, r − 1).Lemma 2.8 ([10], Lemma 3). Let M be a base point free line bundle on X with deg M ≤ g − 1 such that its associated morphism ϕ M is birational.…”