A counterexample to the parity conjecture
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914665357901824 |
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| author | Giovenzana, Franco Giovenzana, Luca Graffeo, Michele Lella, Paolo |
| author_facet | Giovenzana, Franco Giovenzana, Luca Graffeo, Michele Lella, Paolo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_18191 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A counterexample to the parity conjecture Giovenzana, Franco Giovenzana, Luca Graffeo, Michele Lella, Paolo Algebraic Geometry Commutative Algebra 14C05 (Primary) 13D10, 13F55, 13P10 (Secondary) 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. |
| title | A counterexample to the parity conjecture |
| topic | Algebraic Geometry Commutative Algebra 14C05 (Primary) 13D10, 13F55, 13P10 (Secondary) |
| url | https://arxiv.org/abs/2305.18191 |