Complete Linear Series on a Hyperelliptic Curve

In this paper we study complete linear series on a hyperelliptic curve $C$ of arithmetic genus $g$. Let $A$ be the unique line bundle on $C$ such that $|A|$ is a $g^1_2$, and let $\mathcal{L}$ be a line bundle on $C$ of degree $d$. Then $\mathcal{L}$ can be factorized as $\mathcal{L} = A^m \otimes B$ where $m$ is the largest integer satisfying $H^0 (C,\mathcal{L} \otimes A^{-m}) \neq 0$. Let $b = {deg}(B)$. We say that \textit{the factorization type of} $\mathcal{L}$ is $(m,b)$. Our main results in this paper assert that $(m,b)$ gives a precise answer for many natural questions about $\mathcal{L}$.