Base-point-free pencils on triple covers of smooth curves

TL;DR

This paper studies the existence of certain morphisms from smooth algebraic curves to the projective line that do not factor through a given triple covering, generalizing classical results to broader cases.

Contribution

It extends Maroni's classical results on base-point-free pencils from trigonal curves to triple covers of arbitrary smooth irrational curves.

Findings

Identifies conditions for the existence of non-factoring morphisms

Generalizes classical trigonal curve results to triple covers

Provides new insights into the structure of algebraic curve coverings

Abstract

Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$ which do not factor through the covering $f$. For this purpose, we generalize the classical results of Maroni concerning base-point-free pencils on trigonal curves to the case of triple covers of arbitrary smooth irrational curves.