Abstract: |
I will discuss flops of relative dimension one between Gorenstein varieties. It was conjectured and proved in some cases in a joint paper with D.Orlov that the derived categories of flopped varieties are equivalent. For relative dimension one Gorenstein case, it was shown by Van den Bergh using ideas from noncommutative geometry, but his functor is different from the one considered in the above paper. Jointly with A. Bodzenta, we give a new interpretation for both functors and show that one is the inverse of the other. This is based on the L^1 vanishing lemma, which might be of independent interest. We show that the flop-flop functor is the inverse of the spherical twist with respect to the spherical functor given by deriving the null-category of the contraction morphism. We prove that the flop-functor is up to shift the spherical cotwist for another functor, when the base of the contraction is affine. If time permits, I will touch a bit on the related categorical noncommutative deformation theory and schobers. |