A sign that used to annoy me, and still does

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 $(-1{)}^{{\dim }_{\mathbb{C}}{T}_{p}Y}=\sigma =(-1{)}^{\dim Z}$ 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 ${\mathrm{Hilb}}^n({\mathbb{A}}^3)$.

arXiv comment10 pages, comments welcome!