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

Abstract

We give a classification of ordered five points in ${\mathbb{P}}^3$ under the diagonal action of $G{L}_4$ over an algebraically closed field of characteristic $0$, by an explicit description of the diagonal action of $G{L}_4$ 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.