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

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.

This version replaces the previous one, with minor revisions. Accepted for publication in Communications in Algebra (18 pages)