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Fourier-Mukai Transforms in Algebraic Geometry$
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D. Huybrechts

Print publication date: 2006

Print ISBN-13: 9780199296866

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199296866.001.0001

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Flips and Flops

Flips and Flops

(p.246) 11 Flips and Flops
Fourier-Mukai Transforms in Algebraic Geometry

D. Huybrechts

Oxford University Press

This chapter presents a series of results that elucidate the relation between D-equivalence and K-equivalence. Derived categories are first studied. These results can then be used to show that the derived category does not change under the standard flop (Bondal and Orlov), one of the simplest birational correspondences. More recently, Bridgeland was able to show that the derived category is invariant under general three-dimensional flops. The situation is slightly more complicated for a Mukai flop, another classical birational correspondence. In this case, the birational correspondence itself does not define a derived equivalence, but Kawamata and Namikawa were able to prove that one can nevertheless find another Fourier-Mukai kernel that does.

Keywords:   blow-up, D-equivalence, K-equivalence, Mukai flop

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