Configuration spaces of complex and real spheres

TL;DR

This paper investigates the geometric invariant theory quotient of multiple projective spaces under orthogonal group actions, linking it to configuration spaces of spheres through classical isomorphisms.

Contribution

It provides a new interpretation of these quotients as configuration spaces of spheres, connecting algebraic geometry with classical geometry.

Findings

Identifies the GIT-quotient with configuration spaces of spheres.

Establishes a classical isomorphism with the Inversive group.

Provides geometric insights into the structure of these quotient spaces.

Abstract

We study the GIT-quotient of the Cartesian power of projective space modulo the projective orthogonal group. A classical isomorphism of this group with the Inversive group of birational transformations of the projective space of one dimension less allows one to interpret these spaces as configuration spaces of complex or real spheres.