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.
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BibTeX
@misc{booth2026classification,
title = {{On the classification of threefold flops}},
author = {Matt Booth},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24864},
eprint = {2610.24864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24864},
}
RIS
TY - UNPB
TI - On the classification of threefold flops
AU - Matt Booth
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24864
UR - https://arxiv.org/abs/2610.24864
AB - 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.
ER -