On the parity conjecture for Hilbert schemes of points on threefolds

TL;DR

This paper proves the parity conjecture for tangent space dimensions at points parametrizing homogeneous ideals in Hilbert schemes of points on threefolds, extending previous results to a broader class of ideals.

Contribution

It generalizes the parity conjecture to points parametrizing homogeneous ideals and graded modules in Quot schemes of $A^3$, expanding the scope of prior proofs.

Findings

Proves the conjecture for points parametrizing homogeneous ideals.

Extends the conjecture to Quot schemes of $A^3$.

Establishes the conjecture for points parametrizing graded modules.

Abstract

Let $Hilb^d(A^3)$ be the Hilbert scheme of $d$ points in $A^3$, and let $T_z$ denote the tangent space to a point $z \in Hilb^d(A^3)$. Okounkov and Pandharipande have conjectured that $\dim T_z$ and $d$ have the same parity for every $z$. For points $z$ parametrizing monomial ideals, the conjecture was proved by Maulik, Nekrasov, Okounkov, and Pandharipande. In this paper, we settle the conjecture for points $z$ parametrizing homogeneous ideals. In fact, we state a generalization of the conjecture to Quot schemes of $A^3$, and we prove it for points parametrizing graded modules.