A sign that used to annoy me, and still does

• Published in 2023
• Added on 2023-08-14

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 $\mathrm{Hilb}^n(\mathbb A^3)$.

Links

• http://arxiv.org/abs/2306.08457v1
• http://arxiv.org/pdf/2306.08457v1

Other information

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 = {2023-10-09},
	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}
}