Polytopale Konstruktionen in der Algebra

For $n\geq 3$, let $\Omega_n$ be the set of line segments between the vertices of a convex $n$-gon. For $j\geq 2$, a $j$-crossing is a set of $j$ line segments pairwise intersecting in the relative interior of the $n$-gon. For $k\geq 1$, let $\Delta_{n,k}$ be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of $\Omega_n$ not containing any $(k+1)$-crossing. The complex $\Delta_{n,k}$ 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 $\Omega_{2n}$ which can be transformed into each other by a $180^\circ$-rotation of the $2n$-gon. Let $\F_n$ be the set $\Omega_{2n}$ after identification, then the complex $\D_{n,k}$ of type-B generalized triangulations is the simplicial complex of subsets of $\F_n$ not containing any $(k+1)$-crossing in the above sense. For $k = 1$, we have that $\D_{n,1}$ is the simplicial complex of type-B triangulations of the $2n$-gon as defined in \cite{Si} and decomposes into a join of an $(n-1)$-simplex and the boundary of the $n$-dimensional cyclohedron. We demonstrate that $\D_{n,k}$ 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 $\D_{n,k}$. On the algebraical side we give a term-order on the monomials in the variables $X_{ij}, 1\leq i,j\leq n$, such that the corresponding initial ideal of the determinantal ideal generated by the $(k+1)$ times $(k+1)$ minors of the generic $n \times n$ matrix contains the Stanley-Reisner ideal of $\D_{n,k}$. We show that the minors form a Gr\"obner-Basis whenever $k\in\{1,n-2,n-1\}$ 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