Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities

TL;DR

This paper demonstrates that for threefolds with Gorenstein singularities, the moduli space of perverse point sheaves encodes flops and the associated Fourier-Mukai transform provides an equivalence between their derived categories.

Contribution

It establishes a link between Bridgeland's moduli space of perverse point sheaves and flops, proving the Fourier-Mukai transform is an equivalence for these threefolds.

Findings

Bridgeland's moduli space encodes flops for certain threefolds.

Fourier-Mukai transform acts as an equivalence between derived categories.

Provides a categorical understanding of flops in the presence of Gorenstein singularities.

Abstract

The main propose of this paper is to show that Bridgeland's moduli space of perverse point sheaves for certain flopping contractions gives the flops, and the Fourier-Mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories.