On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence

Abstract

Let $G \subset SL_3(\mathbb{C})$ be a finite abelian subgroup, and let $Y = G\operatorname{-}\mathrm{Hilb}(\mathbb{C}^3)$ be the corresponding $G$-Hilbert scheme. Denote by $\Psi: D^b_G(\mathrm{Coh}(\mathbb{C}^3)) \to D^b(\mathrm{Coh}(Y))$ the Bridgeland--King--Reid derived equivalence. For a nontrivial character $\chi$ of $G$, let $\chi^!$ be the corresponding skyscraper sheaf supported at the origin. It is known that $\Psi(\chi^!)$ is always a pure sheaf supported either in degree $0$ or in degree $-1$. We prove that the proportion of characters $\chi$ for which $\Psi(\chi^!)$ is supported in degree $0$ is a rational number lying between $0.25$ and $1$, with both bounds being sharp. Moreover, we exhibit families of resolutions for which these proportions attain certain explicit values within this range.