Fiber polytopes for the projections between cyclic polytopes

Abstract

The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve $(t,t^2,...,t^d):t\in \mathbb{R}$ in ${\mathbb{R}}^d$. For $d^{\prime }>d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $\pi :C(n,d^{\prime })\to C(n,d)$ which "forgets" the last $d^{\prime }-d$ 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 $(n,d,d^{\prime })$ 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.

28 pages with 1 postscript figure

References in corpus (1)

• Lectures on 0/1-polytopes

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