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Reflection groups on smooth projective threefolds
Abstract
Let Y be a smooth projective threefold with H2(Y, 𝒪Y)=0 and unobstructed deformations. We define reflection groups WYsm and WYbig canonically associated to Y, acting on the nef, resp. movable cone of a very general deformation of Y. If −KY is nef, we show that Nef(Y)∩Big(Y) (resp. Mov̅(Y)∩Big(Y)) is a fundamental domain for the action of the corresponding group on the big part of the generic cone. This generalizes work of Wilson and Lutz on Calabi--Yau threefolds. The group WYbig recovers and gives a deformation-theoretic explanation for the Weyl group action on divisor classes of point blowups of ℙ3 studied by Dolgachev--Ortland and Mukai.
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BibTeX
@misc{lutz2026reflection,
title = {{Reflection groups on smooth projective threefolds}},
author = {Wendelin Lutz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32450},
eprint = {2610.32450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32450},
}
RIS
TY - UNPB
TI - Reflection groups on smooth projective threefolds
AU - Wendelin Lutz
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32450
UR - https://arxiv.org/abs/2610.32450
AB - Let Y be a smooth projective threefold with H^2(Y, 𝒪_Y)=0 and unobstructed deformations. We define reflection groups W_Y^sm and W_Y^big canonically associated to Y, acting on the nef, resp. movable cone of a very general deformation of Y. If −K_Y is nef, we show that Nef(Y)∩Big(Y) (resp. Mov̅(Y)∩Big(Y)) is a fundamental domain for the action of the corresponding group on the big part of the generic cone. This generalizes work of Wilson and Lutz on Calabi--Yau threefolds. The group W_Y^big recovers and gives a deformation-theoretic explanation for the Weyl group action on divisor classes of point blowups of ℙ^3 studied by Dolgachev--Ortland and Mukai.
ER -