Smooth and irreducible multigraded Hilbert schemes

TL;DR

This paper proves that multigraded Hilbert schemes for the polynomial ring ZZ[x,y] are smooth and irreducible, confirming a conjecture and advancing understanding of their geometric properties.

Contribution

It establishes the smoothness and irreducibility of multigraded Hilbert schemes for ZZ[x,y], confirming a conjecture by Haiman and Sturmfels.

Findings

Multigraded Hilbert schemes are smooth for ZZ[x,y].

They are also irreducible in this case.

The results confirm a longstanding conjecture.

Abstract

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is ZZ[x,y], which establishes a conjecture of Haiman and Sturmfels.