Description of $GL_3$-orbits on the quadruple projective varieties

TL;DR

This paper classifies and describes the structure of diagonal $GL_3$-orbits on the quadruple product of projective planes, providing explicit representatives and orbit closure relations.

Contribution

It offers the first detailed description of $GL_3$-orbits on $(P^2)^4$, including explicit orbit representatives and their closure relations.

Findings

Explicit representatives for each orbit are provided.

The closure relations among orbits are characterized.

The case illustrates the complexity of orbit structures with infinitely many orbits.

Abstract

This article gives a description of the diagonal $GL_3$-orbits on the quadruple projective variety $(\mathbb P^2)^4$. We give explicit representatives of orbits, and describe the closure relations of orbits. A distinguished feature of our setting is that it is the simplest case where $\mathrm{diag}(GL_n)$ has infinitely many orbits but has an open orbit in the multiple projective space $(\mathbb P^{n-1})^m$.