Complete Linear Series on a Hyperelliptic Curve

TL;DR

This paper investigates the structure of complete linear series on hyperelliptic curves, introducing a factorization approach that precisely characterizes their properties and answers key questions about line bundles.

Contribution

It introduces a factorization method for line bundles on hyperelliptic curves, providing a detailed classification that advances understanding of their linear series.

Findings

Factorization type (m,b) characterizes line bundles on hyperelliptic curves.

Provides explicit answers to natural questions about line bundles.

Enhances understanding of the geometry of hyperelliptic curves.

Abstract

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 $\mathcal{L}$.