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

Let $G\subset S{L}_3(\mathbb{C})$ be a finite abelian subgroup, and let $Y=G\mathrm{-}\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}({\mathbb{C}}^3)$ be the corresponding $G$-Hilbert scheme. Denote by $\mathrm{\Psi }:D_G^b(\mathrm{C}\mathrm{o}\mathrm{h}({\mathbb{C}}^3))\to D^b(\mathrm{C}\mathrm{o}\mathrm{h}(Y))$ 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 $\mathrm{\Psi }({\chi }^{!})$ is always a pure sheaf supported either in degree $0$ or in degree $-1$. We prove that the proportion of characters $\chi$ for which $\mathrm{\Psi }({\chi }^{!})$ 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.