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, t², . . . , t^d) : t ∈ ℝ} in ℝ^d. For d′ > d, we consider the fiber polytope (in the sense of Billera and Sturmfels [6]) associated to the natural projection of cyclic polytopes π : C(n, d′) → 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 π. 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 π 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 π : P → Q where Q has only regular subdivisions and P has two more vertices than its dimension.