ACM embeddings of curves of a quadratic surface (Q2474807)

Let \(C\) be a smooth projective curve, defined over an algebraically closed field of characteristic zero. If \({\mathcal L}\) is a very ample line bundle on \(C\) such that the morphism induced by the linear system \(| {\mathcal L}| \) is a projectively normal embedding then we one says that \(| {\mathcal L}| \) is normally generated. In the present paper the authors study the existence of certain types of normally generated linear systems on \(C\); more specifically, assuming that \(g^1_a\) and \(g^1_b\) are two independent pencils on \(C\), they ask when is \(| s g^1_a + r g^1_b| \) normally generated (where \(r,s \in \mathbb{Z}\)). Among many results, they prove, for instance, the following result. Theorem. Let \(R_1 := \{ (c,d) \in \mathbb{Z}\times\mathbb{Z} \; | \; c \geq a \text{ and } d \leq b - 2\}\), \(R_2 := \{ (c,d) \in \mathbb{Z}\times\mathbb{Z} \; | \; c \leq a - 2 \text{ and } d \geq b\}\) and \(R_3 = \{ (c, d) \in \mathbb{Z}\times\mathbb{Z} \; | \; c \leq a - 2 \text{ and } d \leq b - 2\}\). Then \(| s g^1_a + r g^1_b| \) is normally generated if either \((r, s) \notin R_3\) and no multiple of \((r, s)\) belongs to \(R_1 \cup R_2\) or if \((r, s) \in R_1 \cup R_2\).