Weighted Projective Spaces and a Generalization of Eves' Theorem

TL;DR

This paper generalizes Eves' Theorem to a broader class of point configurations using weighted projective spaces, providing new invariants that distinguish configurations beyond classical methods.

Contribution

It introduces a generalized invariant in weighted projective spaces applicable to more configurations, extending Eves' Theorem.

Findings

The complex invariant can be computed from classical ratios of determinants.

The real invariant can differentiate configurations that classical invariants cannot.

The generalization applies to a larger class of point configurations.

Abstract

For a certain class of configurations of points in space, Eves' Theorem gives a ratio of products of distances that is invariant under projective transformations, generalizing the cross-ratio for four points on a line. We give a generalization of Eves' theorem, which applies to a larger class of configurations and gives an invariant with values in a weighted projective space. We also show how the complex version of the invariant can be determined from classically known ratios of products of determinants, while the real version of the invariant can distinguish between configurations that the classical invariants cannot.