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Let \X be an affine toric variety under a torus \T and let T be a subtorus.The general T-orbit closures in \X and their flat limits are parametrized bythe main component H_0 of the toric Hilbert scheme. Further, 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 andSturmfels. We also describe the relation of H_0 to the main component of theinverse limit of GIT quotients of \X by T.

The main component of the toric Hilbert scheme