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, . . . , td) :t ∈ R } in Rd. Ford′ & d, we consider the fiber polytope (in the sense of Billera and Sturmfels ) 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 mapπ . 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 andP has two more vertices than its dimension.