The largest intersection lattice of a discriminantal arrangement
Let \(\mathcal A\) be an essential arrangement of \(n\) linear hyperplanes in \({\mathbb R}^k\), with normal vectors \(\alpha_1,\ldots,\alpha_n\). The discriminantal arrangement \(B({\mathcal A})\) associated with \(\mathcal A\) is the arrangement in \({\mathbb R}^n\) with normals the nonzero vectors of the form \[ \alpha_S=\sum_{i=1}^{k+1} (-1)^i\det(\alpha_{s_1},\ldots,\hat{\alpha}_{s_i},\ldots,\alpha_{s_{k+1}})e_i, \] where \(S=\{s_1<\cdots<s_{k+1}\}\) runs over all \((k+1)\)-subsets of \([n]\), \(\hat{\mathstrut}\) denoting an omitted vector. The complement of \(B({\mathcal A})\) can be identified with the set of all affine arrangements in \({\mathbb R}^k\) consisting of generic parallel translates of the hyperplanes of \(\mathcal A\). The construction of \(B({\mathcal A})\) is dual to the construction of fiber polytope from a given zonotope. The discriminantal arrangement \(B({\mathcal A})\) was introduced by \textit{Yu. I. Manin} and \textit{V. V. Shekhtman} [Adv. Stud. Pure Math. 17, 289-308 (1989; Zbl 0759.20002)] in the case where \(\mathcal A\) is itself a generic arrangement. Even in this special case the combinatorial type (esp. the intersection lattice) of \(B({\mathcal A})\) is not uniquely determined by that of \({\mathcal A}\). In fact the matroid of \(B({\mathcal A})\) is the dual of the adjoint of that of \({\mathcal A}\) determined by the realization \({\mathcal A}\). Different realizations of the same matroid may yield non-isomorphic adjoints. On the other hand, the combinatorial type of \(B({\mathcal A})\) is constant for \({\mathcal A}\) ranging over a proper Zariski-open set \({\mathcal O}(n,k)\) of ``sufficiently generic'' arrangements. \textit{M. M. Bayer} and \textit{K. A. Brandt} [J. Algebr. Comb. 6, No. 3, 229-246 (1997; Zbl 0884.52013)] conjectured that arrangements in \({\mathcal O}(n,k)\) are very generic, in the sense that their discriminantals have the largest number of flats in each rank among all discriminantal arrangements of essential arrangements of \(n\) hyperplanes in \({\mathbb R}^k\). They defined a poset \(P(n,k)\), whose elements are certain collections of subsets of \([n]\) of size at least \(k+1\), and conjecture that \(P(n,k)\) is isomorphic to the intersection lattice \(L(n,k)\) of \(B({\mathcal A})\) for such sufficiently generic arrangements \({\mathcal A}\). In the paper under review, the author first provides a precise description of the Zariski-open set \({\mathcal O}(n,k)\). It follows from his description that the matroid of any discriminantal arrangement is a weak image of that of \(B({\mathcal A})\) for \({\mathcal A}\) in \({\mathcal O}(n,k)\). Then the theory of weak maps implies that \({\mathcal O}(n,k)\) consists of very generic arrangements. Thus is established the first part of the conjecture of Bayer and Brandt. Next the author observes that the normal vectors \(\alpha_S\), for an arbitrary arrangement, realize the matroid of circuits of the configuration \(\alpha_1,\ldots \alpha_n\), studied by \textit{H. Crapo} in [Adv. Math. 54, 278-301 (1984; Zbl 0593.51008)]. From the description of \({\mathcal O}(n,k)\), it follows from Crapo's work that the matroid of \(B({\mathcal A})\), for \(\mathcal A\) sufficiently generic, is the Dilworth completion \(M(n,k)\) of the \(k\)th lower truncation of the Boolean algebra of rank \(n\). Using Crapo's description of this matroid, the author shows that its lattice of flats, which coincides with \(L(n,k)\), is isomorphic to the Bayer-Brandt poset \(P(n,k)\), thus proving the second part of their conjecture.
scientific article; zbMATH DE number 1460277