Type-B generalized triangulations and determinantal ideals

For n ≥ 3, let Ωn be the set of line segments between the vertices of a convex n-gon. For j ≥ 2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k ≥ 1, let ∆n,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ωn not containing any (k + 1)-crossing. The complex ∆n,k has been the central object of numerous papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line-segments in Ω2n which can be transformed into each other by a 180-rotation of the 2n-gon. Let Fn be the set Ω2n after identification, then the complex Dn,k of type-B generalized triangulations is the simplicial complex of subsets of Fn not containing any (k + 1)-crossing in the above sense. For k = 1, we have that Dn,1 is the simplicial complex of type-B triangulations of the 2n-gon as defined in [Si] and decomposes into a join of an (n−1)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that Dn,k is a pure, k(n − k)− 1 + kn dimensional complex that decomposes into a kn− 1-simplex and a k(n− k)− 1 dimensional homology sphere. For k = n − 2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of Dn,k. On the algebraical side we give a term-order on the monomials in the variables Xij , 1 ≤ i, j ≤ n, such that the corresponding initial ideal of the determinantal ideal generated by the (k + 1) times (k + 1) minors of the generic n × n matrix contains the Stanley-Reisner ideal of Dn,k. We show that the minors form a Gröbner-Basis whenever k ∈ {1, n − 2, n − 1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k < n.