### Abstract

For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_\theta, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter \theta. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_\theta for some parameter \theta. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also uncover explicit equivalences between the derived categories of moduli M_\theta for parameters lying in adjacent GIT chambers.

Original language | English |
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Pages (from-to) | 259-307 |

Journal | Duke Mathematical Journal |

Volume | 124 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2004 |

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## Cite this

Craw, A., & Ishii, A. (2004). Flops of

*G*-Hilb and equivalences of derived categories by variation of GIT quotient.*Duke Mathematical Journal*,*124*(2), 259-307. https://doi.org/10.1215/S0012-7094-04-12422-4