Tohoku Mathematical Journal
2008
September
SECOND SERIES VOL. 60, NO. 3

Tohoku Math. J.
60 (2008), 365-382

Title THE MAIN COMPONENT OF THE TORIC HILBERT SCHEME

Author Olga V. Chuvashova

(Received April 10, 2006, revised January 7, 2008)
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 $T$.

2000 Mathematics Subject Classification. Primary 14C05; Secondary 52B20, 14M25.
Key words and phrases. Toric Hilbert scheme, fiber polytope, toric Chow quotient.

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