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arXiv · 2305.18191

A counterexample to the parity conjecture

Franco Giovenzana·Luca Giovenzana·Michele Graffeo·Paolo Lella
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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.

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Franco Giovenzana, Luca Giovenzana, Michele Graffeo, Paolo Lella. 2023-05-29. A counterexample to the parity conjecture. https://arxiv.org/abs/2305.18191

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