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Borel subgroups of the automorphism group of an affine variety
Abstract
We study Borel subgroups of the automorphism group of an affine variety X and their associated Lie algebras. We prove a bijective correspondence between the set of Borel subgroups of Aut(X) and the set of maximal nested unipotent subgroups of Aut(X). We show that if X is factorial of dimension at most 3, then the Lie algebra of every maximal nested unipotent subgroup has the special property of being what we call "integral"; we also show that this fails for X = affine space of dimension > 3. As an application of our results, we show that Nagata's automorphism is contained in a unique Borel subgroup.
Related
- Presentation of Borel subgroups of Kac-Moody groups over local rings
- Quivers variety for affine orthogonal quiver
- Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
- Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two
- Neutral components of automorphism groups of (semi)rigid affine varieties
BibTeX
@misc{chitayat2026borel,
title = {{Borel subgroups of the automorphism group of an affine variety}},
author = {Michael Chitayat and Daniel Daigle and Andriy Regeta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06664},
eprint = {2610.06664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06664},
}
RIS
TY - UNPB
TI - Borel subgroups of the automorphism group of an affine variety
AU - Michael Chitayat
AU - Daniel Daigle
AU - Andriy Regeta
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06664
UR - https://arxiv.org/abs/2610.06664
AB - We study Borel subgroups of the automorphism group of an affine variety X and their associated Lie algebras. We prove a bijective correspondence between the set of Borel subgroups of Aut(X) and the set of maximal nested unipotent subgroups of Aut(X). We show that if X is factorial of dimension at most 3, then the Lie algebra of every maximal nested unipotent subgroup has the special property of being what we call "integral"; we also show that this fails for X = affine space of dimension > 3. As an application of our results, we show that Nagata's automorphism is contained in a unique Borel subgroup.
ER -