THE MAIN COMPONENT OF THE TORIC HILBERT SCHEME

概要

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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 $T$.