### Abstract

Given an irreducible affine algebraic variety X of dimension n≥2, we let SAut(X) denote the special automorphism group of X, that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X_{reg}, then it is infinitely transitive on X_{reg}. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x∈X_{reg}the tangent space T_{x}X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We also provide various modifications and applications.

Original language | English (US) |
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Pages (from-to) | 767-823 |

Number of pages | 57 |

Journal | Duke Mathematical Journal |

Volume | 162 |

Issue number | 4 |

DOIs | |

State | Published - Mar 1 2013 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*162*(4), 767-823. https://doi.org/10.1215/00127094-2080132

**Flexible varieties and automorphism groups.** / Arzhantsev, I.; Flenner, H.; Kaliman, Shulim; Kutzschebauch, F.; Zaidenberg, M.

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 162, no. 4, pp. 767-823. https://doi.org/10.1215/00127094-2080132

}

TY - JOUR

T1 - Flexible varieties and automorphism groups

AU - Arzhantsev, I.

AU - Flenner, H.

AU - Kaliman, Shulim

AU - Kutzschebauch, F.

AU - Zaidenberg, M.

PY - 2013/3/1

Y1 - 2013/3/1

N2 - Given an irreducible affine algebraic variety X of dimension n≥2, we let SAut(X) denote the special automorphism group of X, that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus Xreg, then it is infinitely transitive on Xreg. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x∈Xregthe tangent space TxX is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We also provide various modifications and applications.

AB - Given an irreducible affine algebraic variety X of dimension n≥2, we let SAut(X) denote the special automorphism group of X, that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus Xreg, then it is infinitely transitive on Xreg. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x∈Xregthe tangent space TxX is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We also provide various modifications and applications.

UR - http://www.scopus.com/inward/record.url?scp=84877146397&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877146397&partnerID=8YFLogxK

U2 - 10.1215/00127094-2080132

DO - 10.1215/00127094-2080132

M3 - Article

AN - SCOPUS:84877146397

VL - 162

SP - 767

EP - 823

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 4

ER -