The moduli space of \((3,3,3)\) trilinear forms
In a previous paper [The classification of (3,3,3) trilinear forms, J. Reine Angew. Math. 468, 49-75 (1995; Zbl 0858.11023)], we described the set of \(G\)-orbits of smooth cuboids and derived explicit matrix representations for singular cuboids. In this paper, we use the matrix representations of singular cuboids to investigate the relations among the orbits. Let \(U,V\) and \(W\) be three dimensional vector spaces over \(\mathbb{C}\) (or an algebraically closed field with characteristic not equal to 2 or 3). We prove that the moduli space of trilinear forms on \(U^* \otimes V^* \otimes W^*\) is isomorphic to \(\mathbb{P}^2\) by applying geometric invariant theory to the action of \(PGL(U) \times PGL(V) \times PGL(W)\) on \(\mathbb{P} (U\otimes V \otimes W)\).
matrix representations of singular cuboids
scientific article; zbMATH DE number 866153
scientific article