ACM embeddings of curves of a quadric surface

Resum

Let $C$ be a smooth integral projective curve admitting two pencils $g^1_a$ and $g^1_b$ such that $g^1_a + g^1_b$ is birational. We give conditions in order that the complete linear system $\vert sg^1_a + rg^1_b\vert$ be normally generated or very ample.