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Neutral components of automorphism groups of (semi)rigid affine varieties
Abstract
We say that an affine variety X is (semi)rigid if all nontrivial actions of the additive group on X have the same invariant ring. Then all such actions comprise the abelian subgroup denoted SAut(X). We prove that X is (semi)rigid precisely when the neutral component Aut∘(X) of its automorphism group is nested. In this case, the neutral component is the semidirect product of any maximal algebraic torus and SAut(X). The proof uses Laurent expansions of algebraic curves in Aut∘(X).
Related
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BibTeX
@misc{perepechko2026neutral,
title = {{Neutral components of automorphism groups of (semi)rigid affine varieties}},
author = {Alexander Perepechko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10401},
eprint = {2609.10401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10401},
}
RIS
TY - UNPB
TI - Neutral components of automorphism groups of (semi)rigid affine varieties
AU - Alexander Perepechko
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10401
UR - https://arxiv.org/abs/2609.10401
AB - We say that an affine variety X is (semi)rigid if all nontrivial actions of the additive group on X have the same invariant ring. Then all such actions comprise the abelian subgroup denoted SAut(X). We prove that X is (semi)rigid precisely when the neutral component Aut^∘(X) of its automorphism group is nested. In this case, the neutral component is the semidirect product of any maximal algebraic torus and SAut(X). The proof uses Laurent expansions of algebraic curves in Aut^∘(X).
ER -