The main component of the toric Hilbert scheme

Abstract

Let $\boldsymbol{X}$ be an affine toric variety with big torus $\boldsymbol{T}\subset \boldsymbol{X}$ and let $T\subset\boldsymbol{T}$ be a subtorus. The general $T$-orbit closures in $\boldsymbol{X}$ and their flat limits are parametrized by the main component $H_0$ of the toric Hilbert scheme. Further, the quotient torus $\boldsymbol{T}/T$ 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 of $\boldsymbol{X}$ by $T$.