Non-connected toric Hilbert schemes

We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc., 13:3 (2000), 611--637 not only in size, but also in that their toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.Comment: 18 pages, 2 figures. Except for Remark 2.6 (see below) changes w.r.t. version 2 are mostly minor editings suggested by an anonimous referee of "Mathematische Annalen". The paper has been accepted in that journal. Most of the contents of Remark 2.6 have been deleted, since there was a flaw in the argumen

On \pi-surfaces of four-dimensional parallelohedra

We show that every four-dimensional parallelohedron P satisfies a recently found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi conjecture being true for P. Namely we show that for every four-dimensional parallelohedron P the group of rational first homologies of its \pi-surface is generated by half-belt cycles.Comment: 16 pages, 7 figure

Periodic Tilings and Auxetic Deployments

We investigate geometric characteristics of a specific planar periodic framework with three degrees of freedom. While several avatars of this structural design have been considered in materials science under the name of chiral or missing rib models, all previous studies have addressed only local properties and limited deployment scenarios. We describe the global configuration space of the framework and emphasize the geometric underpinnings of auxetic deformations. Analogous structures may be considered in arbitrary dimension

Reflection groups and polytopes over finite fields, II

When the standard representation of a crystallographic Coxeter group $\Gamma$ is reduced modulo an odd prime $p$, a finite representation in some orthogonal space over $\mathbb{Z}_p$ is obtained. If $\Gamma$ has a string diagram, the latter group will often be the automorphism group of a finite regular polytope. In Part I we described the basics of this construction and enumerated the polytopes associated with the groups of rank 3 and the groups of spherical or Euclidean type. In this paper, we investigate such families of polytopes for more general choices of $\Gamma$, including all groups of rank 4. In particular, we study in depth the interplay between their geometric properties and the algebraic structure of the corresponding finite orthogonal group.Comment: 30 pages (Advances in Applied Mathematics, to appear

Fiber polytopes for the projections between cyclic polytopes

The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve ${(t,t^2,...,t^d):t \in \reals}$ in $\reals^d$. For $d' >d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $\pi: C(n,d') \to C(n,d)$ which "forgets" the last $d'-d$ coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of $C(n,d)$ which are induced by the map $\pi$. Our main result characterizes the triples $(n,d,d')$ for which the fiber polytope is canonical in either of the following two senses: - all polytopal subdivisions induced by $\pi$ are coherent, - the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection $\pi:P\to Q$ where $Q$ has only regular subdivisions and $P$ has two more vertices than its dimension.Comment: 28 pages with 1 postscript figur