On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence
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| Format: | Preprint |
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2025
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| _version_ | 1866914047196135424 |
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| author | Tsvelikhovskiy, Boris |
| author_facet | Tsvelikhovskiy, Boris |
| contents | 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 $Ψ: D^b_G(\mathrm{Coh}(\mathbb{C}^3)) \to D^b(\mathrm{Coh}(Y))$ the Bridgeland--King--Reid derived equivalence. For a nontrivial character $χ$ of $G$, let $χ^!$ be the corresponding skyscraper sheaf supported at the origin. It is known that $Ψ(χ^!)$ is always a pure sheaf supported either in degree $0$ or in degree $-1$. We prove that the proportion of characters $χ$ for which $Ψ(χ^!)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_15825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence Tsvelikhovskiy, Boris Algebraic Geometry Representation Theory 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 $Ψ: D^b_G(\mathrm{Coh}(\mathbb{C}^3)) \to D^b(\mathrm{Coh}(Y))$ the Bridgeland--King--Reid derived equivalence. For a nontrivial character $χ$ of $G$, let $χ^!$ be the corresponding skyscraper sheaf supported at the origin. It is known that $Ψ(χ^!)$ is always a pure sheaf supported either in degree $0$ or in degree $-1$. We prove that the proportion of characters $χ$ for which $Ψ(χ^!)$ 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. |
| title | On a Dichotomy for Skyscraper Sheaves under the Bridgeland-King-Reid Equivalence |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2509.15825 |