The main component of the Toric Hilbert scheme

Abstract

Let X be an affine toric variety with big torus T ⊂ X and let ⊂ C T be a subtorus. The general T-orbit closures in X and their flat limits are parametrized by the main component H0 of the toric Hilbert scheme. Further, the quotient torus T/T acts on H0 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 H0 to the main component of the inverse limit of GIT quotients of X by T.