A Shorter Proof of Marten's Theorem

TL;DR

This paper presents a simplified proof of Marten's theorem for smooth curves, utilizing Hopf's theorem to streamline the argument and enhance the theorem's conclusions.

Contribution

It offers a new, more straightforward proof of Marten's theorem that also strengthens its results using Hopf's theorem.

Findings

Simplified proof of Marten's theorem

Characterization of equality case as hyper-elliptic curves

Enhanced conclusions about the theorem's implications

Abstract

This note is intended to give a new proof on Marten's theorem stating that $\dim(\wrd) \leq d-2r$ for any smooth curve with equality occurring exactly in the case when C is hyper-elliptic. The proof follows the general lines of the proof given in 'The Geometry of Algebraic Curves'. What is different is that it uses Hopf's theorem to simplify the proof and strengthen the conclusions of the theorem.