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 $T$.
@article{1223057734,
author = {Chuvashova, Olga V.},
title = {The main component of the toric Hilbert scheme},
journal = {Tohoku Math. J. (2)},
volume = {60},
number = {1},
year = {2008},
pages = { 365-382},
language = {en},
url = {http://dml.mathdoc.fr/item/1223057734}
}
Chuvashova, Olga V. The main component of the toric Hilbert scheme. Tohoku Math. J. (2), Tome 60 (2008) no. 1, pp. 365-382. http://gdmltest.u-ga.fr/item/1223057734/