On the classification of threefold flops

Abstract

We show that possibly singular threefold flopping contractions are completely classified by their associated derived contraction algebras, considered up to derived Morita equivalence; in contrast to the smooth case the extra derived structure is necessary in general. This extends previous work of the author on simple flops, and follows from a more general theorem about isolated hypersurface singularities. The key new input is to show that derived localisation at a Hochschild class is Morita invariant; this result raises some interesting questions related to the classification of recollements. We also show that derived contraction algebras admit a deformation-theoretic interpretation in families in a general setting that encompasses threefold flops, flips, and divisor-to-curve contractions.