Unstable hyperplanes for Steiner bundles and multidimensional matrices (Q2726408)

The authors study some properties of the natural action of \(\text{SL} (V_0)\times \cdots\times \text{SL}(V_p)\) on nondegenerate multidimensional complex matrices \(A\in P(V_0 \otimes\cdots \otimes V_p)\) of boundary formate; in particular they characterize the nonstable ones as the matrices which are in the orbit of a ``triangular'' matrix, and the matrices with a stabilizer containing \(\mathbb{C}^*\) as those which are in the orbit of a ``diagonal'' matrix. For \(p=2\) it turns out that a non-degenerate matrix \(A\in P(V_0 \otimes V_1\otimes V_2)\) detects a Steiner bundle \(S_A\) on the projective space \(\mathbb{P}^n\), \(n=\dim (V_2)-1\). The authors prove that the symmetry group of a Steiner bundle is contained in SL(2) and that the SL(2)-invariant Steiner bundles are exactly the bundles introduced by Schwarzenberger, which correspond to ``identity'' matrices. The authors characterize the points of the moduli space of Steiner bundles which are stable for the action of \(\Aut (\mathbb{P}^n)\). In the opposite direction they obtain some results about Steiner bundles which imply properties of matrices. Finally, the Gale transform for Steiner bundles introduced by Dolgachev and Kapanov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.NEWLINENEWLINENEWLINEThe paper contains the following sections: Multidimensional matrices of boundary format and geometric invariant theory; Preliminaries about Steiner bundles; The Schwarzenberger bundles; A filtration of \(\varphi_{n,k}\) and the Gale transform on Steiner bundles; Moduli spaces of Steiner bundles and geometric invariant theory.