A counterexample to the parity conjecture

TL;DR

This paper presents a counterexample to the parity conjecture for the Hilbert scheme of points in three-dimensional affine space, showing the conjecture does not hold in the general non-homogeneous case.

Contribution

The authors construct a specific zero-dimensional scheme in ext{Hilb}^{12} ext{A}^3 that disproves the parity conjecture in the non-homogeneous setting.

Findings

Counterexample in ext{Hilb}^{12} ext{A}^3

Disproves the parity conjecture in the general case

Shows conjecture holds only for special classes like monomial and homogeneous ideals

Abstract

Let $[Z]\in\text{Hilb}^d \mathbb A^3$ be a zero-dimensional subscheme of the affine three-dimensional complex space of length $d>0$. Okounkov and Pandharipande have conjectured that the dimension of the tangent space of $\text{Hilb}^d \mathbb A^3$ at $[Z]$ and $d$ have the same parity. The conjecture was proven by Maulik, Nekrasov, Okounkov and Pandharipande for points $[Z]$ defined by monomial ideals and very recently by Ramkumar and Sammartano for homogeneous ideals. In this paper we exhibit a family of zero-dimensional schemes in $\text{Hilb}^{12} \mathbb A^3$, which disproves the conjecture in the general non-homogeneous case.