Configuration of five points in $\mathbb P^3$ and their limits

TL;DR

This paper classifies configurations of five points in projective 3-space under the action of the general linear group, providing explicit descriptions and orbit closure relations in a complex geometric setting.

Contribution

It offers a detailed classification of five-point configurations in ^3 under GL_4 action, including explicit orbit descriptions and closure relations, in a novel geometric context.

Findings

Explicit classification of five-point configurations in ^3.

Description of orbit closure relations among infinitely many orbits.

Analysis of group actions on projective varieties in a new setting.

Abstract

We give a classification of ordered five points in $\mathbb P^3$ under the diagonal action of $GL_4$ over an algebraically closed field of characteristic $0$, by an explicit description of the diagonal action of $GL_4$ on the quintuple of the projective varieties $\mathbb P^3$. This is the second simplest setting, where a reductive subgroup of $H$ of $G$ has an open orbit in a (generalised) flag variety $X$ of $G$ but $\#(H\backslash X)=\infty$. The closure relations among infinitely many orbits are also given.