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arXiv · 2306.08457

A sign that used to annoy me, and still does

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

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Andrea T. Ricolfi. 2023-06-14. A sign that used to annoy me, and still does. https://arxiv.org/abs/2306.08457

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