Toric Hilbert schemes

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Higher-dimensional analogues of stable curves (2006, background): “We note that there exists another moduli space closely related to our moduli space M of stable toric varieties: it is the toric Hilbert scheme HilbTPn [46], [24] parameterizing subschemesX ⊂ P corresponding to the multiplicity-free multigraded algebras.”
Stable reductive varieties II: projective case (2002, background): “d the image is its “shadow cycle”. The works [KSZ92, GKZ94] describe the toric Chow variety parameterizing these cycles. Another related moduli space is the toric Hilbert scheme of Peeva and Stillman [PS02]. In addition to the work in the toric case, a starting point for our investigation was M. Kap”
Polyhedral divisors and algebraic torus actions (2003, background): “ toricvariety associated to the state polytope of P. For the notion of P- (or usually called A-) graded ideals and their base spaces, the toric Hilbert schemes, we refer for example to [18], [20] and [21]. Proof of Proposition 7.2. First note that OY-algebra Ais free and hence ﬂat, because locally, ”
Existence of unimodular triangulations — positive results (2014, methodology): “The toric Hilbert scheme, as introduced by Peeva and Stillman [92], is the set of all A-graded ideals with a suitable algebraic structure defined by some determinantal equations.”