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