  Robertson (1988) suggested a model for the realization space of a convex
d-dimensional polytope and an approach via the implicit function theorem, which
-- in the case of a full rank Jacobian -- proves that the realization space is
a manifold of dimension NG(P):=d(f_0+f_{d-1})-f_{0,d-1}, which is the natural
guess for the dimension given by the number of variables minus the number of
quadratic equations that are used in the definition of the realization space.
While this indeed holds for many natural classes of polytopes (including simple
and simplicial polytopes, as well as all polytopes of dimension at most 3),and
Robertson claimed this to be true for all polytopes, Mnev's (1986/1988)
Universality Theorem implies that it is not true in general: Indeed, (1) the
centered realization space is not a smoothly embedded manifold in general, and
(2) it does not have the dimension NG(P) in general.
  In this paper we develop Jacobian criteria for the analysis of realization
spaces. From these we get easily that for various large and natural classes of
polytopes the realization spaces are indeed manifolds, whose dimensions are
given by NG(P). However, we also identify the smallest polytopes where the
dimension count (2) and thus Robertson's claim fails, among them the bipyramid
over a triangular prism. For the property (1), we analyze the classical
24-cell: We show that the realization space has at least the dimension 48, and
it has points where it is a manifold of this dimension, but it is not smoothly
embedded as a manifold everywhere.
