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

Abstract

Let $G\subset S{L}_3(\mathbb{C})$ be a finite abelian subgroup, and let $Y=G-\mathrm{Hilb}({\mathbb{C}}^3)$ be the corresponding $G$-Hilbert scheme. Denote by $\Psi :D_G^b(\mathrm{Coh}({\mathbb{C}}^3))\to D^b(\mathrm{Coh}(Y))$ 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.

arXiv admin note: substantial text overlap with arXiv:2201.07847