New and General Type Meromorphic $1$-forms on Curves

TL;DR

This paper constructs a broad family of new meromorphic 1-forms on various curves and links this to the Hurwitz problem, offering an algorithm to identify new forms among those on the Riemann Sphere.

Contribution

It introduces explicit constructions of new meromorphic 1-forms on different curves and connects these to the Hurwitz realization problem, providing a novel algorithmic approach.

Findings

Constructed large families of new meromorphic 1-forms on curves.

Established a connection to the Hurwitz realization problem.

Provided an algorithm to distinguish new forms on the Riemann Sphere.

Abstract

In this article, we study the existence of new and general type meromorphic $1$-forms on curves through explicit construction. Specifically, we have constructed a large family of new and general type meromorphic $1$-forms on $\mathbb{P}^1,$ elliptic and hyperelliptic curves. We also established a connection to the Hurwitz realization problem of branch cover for the Riemann Sphere, which provides an algorithm to determine whether a $1$-form on $\mathbb{P}^1$ (of some restricted class) is new or old.