The toric Hilbert scheme of a rank two lattice is smooth and irreducible
Abstract
The toric Hilbert scheme of a lattice L in Zn is the multigraded Hilbert scheme parameterizing all ideals in k[x1,...,xn] with Hilbert function value one for every degree in the grading monoid Nn/L. In this paper we show that if L is two-dimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert scheme of a rank three lattice can be reducible.
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