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 σ∈{±1}, then all of its components Z⊂Y are generically reduced and satisfy (−1)dimC⁡TpY=σ=(−1)dim⁡Z for p∈Z a general point. Given the recent counterexamples to the parity conjecture for the Hilbert scheme of points Hilbn(A3), our argument suggests a possible path to disprove the constancy of the Behrend function of Hilbn(A3).