Gregory G. Smith: Smooth and irreducible multigraded Hilbert schemes

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. In this talk, we will show that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y]; this establishes a conjecture of Haiman and Sturmfels.