Unstable hyperplanes for Steiner bundles and multidimensional matrices

Abstract

We study some properties of the natural action of $SL(V_0)\times ...\times SL(V_p)$ on nondegenerate multidimensional complex matrices $A\in \P (V_0\otimes ...\otimes V_p)$ of boundary format(in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the non stable ones,as the matrices which are in the orbit of a "triangular" matrix, and the matrices with a stabilizer containing $\C^*$, 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 $S_A$ (in the sense of Dolgachev and Kapranov) on the projective space ${\P }^n,n=dim(V_2)-1$. As a consequence we 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 [Schw], which correspond to "identity" matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of $Aut({\P }^n)$, answering in the first nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes of $S_A$ (counting multiplicities) produces an interesting discrete invariant of $A$, which can take the values $0,1,2,...,\dim V_0+1$ or $\infty$; the $\infty$ case occurs if and only if $S_A$ is Schwarzenberger (and $A$ is an identity). Finally, the Gale transform for Steiner bundles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.

27 pages, plain tex, a missing case in theorem 2.6 is added

Cited by in corpus (15)

• Hyperplane arrangements of Torelli type
• Schwarzenberger bundles on smooth projective varieties
• Logarithmic Bundles Of Hypersurface Arrangements In P^n
• Schwarzenberger bundles of arbitary rank on the projective space
• The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space
• Nondegenerate multidimensional matrices and instanton bundles
• Logarithmic sheaves attached to arrangements of hyperplanes
• Jumping pairs of Steiner bundles
• Group Actions on Vector Bundles on the Projective Plane
• On the moduli space of the Schwarzenberger bundles
• On a compactification of the moduli space of the rational normal curves
• Invariant multidimensional matrices
• Group action on instanton bundles over $\PP^3$
• Logarithmic bundles of multi-degree arrangements in ${\mathbf{P}}^n$
• Stabilizers for nondegenerate matrices of boundary format and Steiner bundles