The toric Hilbert scheme of a rank two lattice is smooth and irreducible

Abstract

The toric Hilbert scheme of a lattice L⊆ Z n is the multigraded Hilbert scheme parameterizing all homogeneous ideals I in S= k [x 1,…,x n] such that the Hilbert function of the quotient S/ I has value one for every g in the grading monoid G += N n/ 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.