# A sign that used to annoy me, and still does

- Published in 2023
- Added on

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We provide a proof of the following fact: if a complex scheme $Y$ has Behrend function constantly equal to a sign $\sigma \in \{\pm 1\}$, then all of its components $Z \subset Y$ are generically reduced and satisfy $(-1)^{\mathrm{dim}_{\mathbb C} T_pY} = \sigma = (-1)^{\mathrm{dim}Z}$ for $p \in Z$ a general point. Given the recent counterexamples to the parity conjecture for the Hilbert scheme of points $\mathrm{Hilb}^n(\mathbb A^3)$, our argument suggests a possible path to disprove the constancy of the Behrend function of $\mathrm{Hilb}^n(\mathbb A^3)$.

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- key
- Asignthatusedtoannoymeandstilldoes
- type
- article
- date_added
- 2023-08-14
- date_published
- 2023-08-29

### BibTeX entry

@article{Asignthatusedtoannoymeandstilldoes, key = {Asignthatusedtoannoymeandstilldoes}, type = {article}, title = {A sign that used to annoy me, and still does}, author = {Andrea T. Ricolfi}, abstract = {We provide a proof of the following fact: if a complex scheme {\$}Y{\$} has Behrend function constantly equal to a sign {\$}\sigma \in \{\{}\pm 1\{\}}{\$}, then all of its components {\$}Z \subset Y{\$} are generically reduced and satisfy {\$}(-1)^{\{}\mathrm{\{}dim{\}}{\_}{\{}\mathbb C{\}} T{\_}pY{\}} = \sigma = (-1)^{\{}\mathrm{\{}dim{\}}Z{\}}{\$} for {\$}p \in Z{\$} a general point. Given the recent counterexamples to the parity conjecture for the Hilbert scheme of points {\$}\mathrm{\{}Hilb{\}}^n(\mathbb A^3){\$}, our argument suggests a possible path to disprove the constancy of the Behrend function of {\$}\mathrm{\{}Hilb{\}}^n(\mathbb A^3){\$}.}, comment = {}, date_added = {2023-08-14}, date_published = {2023-08-29}, urls = {http://arxiv.org/abs/2306.08457v1,http://arxiv.org/pdf/2306.08457v1}, collections = {attention-grabbing-titles}, url = {http://arxiv.org/abs/2306.08457v1 http://arxiv.org/pdf/2306.08457v1}, year = 2023, urldate = {2023-08-14}, archivePrefix = {arXiv}, eprint = {2306.08457}, primaryClass = {math.AG} }