Normal generation of line bundles on multiple coverings

TL;DR

This paper investigates conditions under which line bundles on multiple coverings of algebraic curves are normally generated, extending known results for line bundles of degree at least 2g+1.

Contribution

It provides new criteria for the normal generation of line bundles on multiple coverings, broadening understanding beyond classical degree thresholds.

Findings

Conditions for normal generation of line bundles on multiple coverings

Identification of line bundles failing to be normally generated

Extension of classical results to more general coverings

Abstract

Any line bundle $\cl $ on a smooth curve $C$ of genus $g$ with $\deg \cl \ge 2g+1$ is normally generated, i.e., $\varphi_\cl (C)\subseteq \mathbb P H^0 (C,\cl)$ is projectively normal. However, it has known that more various line bundles of degree $d$ failing to be normally generated appear on multiple coverings of genus $g$ as $d$ becomes smaller than $2g+1$. Thus, investigating the normal generation of line bundles on multiple coverings can be an effective approach to the normal generation. In this paper, we obtain conditions for line bundles on multiple coverings being normally generated or not, respectively.