Stabilizers for nondegenerate matrices of boundary format and Steiner bundles (Q1884148)

In this paper nondegenerate multidimensional matrices of boundary format in \(V_0\otimes\cdots\otimes V_p\) are investigated by their link with Steiner vector bundles on product of projection spaces. The main new technique introduced in this paper is the use of jumping hyperplanes for bundles on the product of \((p- 1)\) projective spaces. As the main result, the stabilizer under the action of \(\text{SL} (V_0) \times \text{SL} (V_1) \times \dots \times \text{SL} (V_p)\) is completely described (in Section 2: Jumping hyperplanes and stabilizers; Stab \((A)^0\) denotes the connected component of the identity). Theorem. Assume that a hypermatrix \(A \in {\mathbb P} (V_0 \otimes V_1 \otimes \dots \otimes V_p)\) is nondegenerate in the sense of Gel'fand-Kapranov-Zelevinsky's theory of multidimensional determinants. Then there exists a two-dimensional complex vector space \(U\) such that \(\text{Stab} (A)^0 \subseteq \text{SL} (U)\). Moreover, \(\text{Stab} (A)^0\) is either \(0\), \(\mathbb C\), \({\mathbb C}^{*}\), or \(\text{SL} (2)\). The last case occurs if and only if \(A\) is an identity hypermatrix. This extends a result of \textit{V. Ancona} and \textit{G. Ottaviani} [Adv. Geom. 1, No.2, 165--192 (2001; Zbl 0983.14034)].