Computing heights on weighted projective spaces

TL;DR

This paper extends the concept of height to weighted projective spaces, providing methods for computation and demonstrating applications to hyperelliptic curves and weighted moduli spaces.

Contribution

It introduces the weighted height on weighted projective spaces, establishing its properties and computational techniques, and applies it to study hyperelliptic curves over Q.

Findings

Weighted height can be computed explicitly.

Properties of the weighted height are established.

Applications to hyperelliptic curves and moduli spaces are demonstrated.

Abstract

In this note we extend the concept height on projective spaces to that of weighted height on weighted projective spaces and show how such a height can be computed. We prove some of the basic properties of the weighted height and show how it can be used to study hyperelliptic curves over Q. Some examples are provided from the weighted moduli space of binary sextics and octavics.