The main component of the toric Hilbert scheme.

Let X be an affine toric variety under a torus T and let T' be a
subtorus. The generic T'-orbit closures in X and their flat limits are parametrized by the main component H_0 of the toric Hilbert scheme (whose existence follows from work of Haiman and Sturmfels). Futher, the quotient torus T/T' acts on H_0 with a dense orbit. We describe the fan of this toric variety; this leads us to an integral analogue of the fiber polytope of Billera and Sturmfels. We also describe the relation of H_0 to the main component of the inverse limit of GIT-quotients.