Smooth and irreducible multigraded Hilbert schemes

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by all abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a Conjecture of Haiman and Sturmfels. (C) 2009 Gregory G. Smith. Published by Elsevier Inc. All rights reserved