Derivation modules of orthogonal duals of hyperplane arrangements

Kung, Joseph P.S. ; Schenck, Hal

Journal of Algebraic Combinatorics, Tome 24 (2006) no. 3, pp. 253-262.

Résumé

Summary: Let $A$ be an n $\times d$ matrix having full rank $n$. An orthogonal dual A $^{\perp }$ of $A$ is a (d-n) $\times d$ matrix of rank $( d - n)$ such that every row of A $^{\perp }$ is orthogonal (under the usual dot product) to every row of $A$. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of $d$ hyperplanes in $n$-dimensional space with the n $\times d$ matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. When n $\geq 5$, we show that if the matroid (or the lattice of intersection) of an $n$-dimensional essential arrangement $A {\mathcal A}$ contains a modular copoint whose complement spans, then the derivation module of the orthogonally dual arrangement $A {\mathcal A} ^{\perp }$ has projective dimension at least $\lceil n(n+2)/4 \rceil - 3$.

Détail

Keywords: keywords hyperplane arrangement, module of derivations, projective dimension, matroid, orthogonal duality