The genus of a curve of Fermat type

TL;DR

This paper calculates the genus of Fermat-type curves on weighted projective planes by analyzing their local properties, smoothness, and ramification, providing a method to determine their topological complexity.

Contribution

It introduces a new approach to compute the genus of Fermat-type curves on weighted projective planes using desingularization and Hurwitz's theorem.

Findings

Curves are shown to be smooth after desingularization.

A specific map to ${\\mathbb P^1}$ is constructed with known ramification.

The genus of the curves is explicitly determined.

Abstract

In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to the one from Fermat's last theorem. We begin by finding local coordinates for the standard affine cover of the plane, and then prove that the curve is smooth. This is done by pulling the curve up to the surface's desingularization. Then a map from the curve to ${\mathbb P^1}$ is constructed, and it's ramification divisor is determined. We conclude by applying Hurwitz's theorem to this map to obtain $C$'s genus.