Description of $G{L}_3$-orbits on the quadruple projective varieties

Abstract

This article gives a description of the diagonal $G{L}_3$-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 $\mathrm{diag}(G{L}_n)$ has infinitely many orbits but has an open orbit in the multiple projective space $({\mathbb{P}}^{n-1}{)}^m$.