Enumerative geometry of the curves defined by y^{d}=f(x)

TL;DR

This paper investigates algebraic plane curves as coverings of the projective line, employing combinatorial methods to enumerate branched coverings, with a focus on curves defined by y^{d}=f(x) over arbitrary fields.

Contribution

It introduces a combinatorial approach to enumerate branched coverings of the projective line by algebraic curves defined by y^{d}=f(x), applicable over fields of any characteristic.

Findings

Developed a combinatorial enumeration method for branched coverings

Applied the method to curves defined by y^{d}=f(x)

Extended results to arbitrary characteristic fields

Abstract

We study plane algebraic curves defined over a field k of arbitrary characteristic as coverings of the the projective line and the problem of enumerating branched coverings of $\mathbb{P}^{1}$ by using combinatorial methods.