On the uniqueness of ℂ*-actions on affine surfaces

Flenner, Hubert; Zaidenberg, Mikhail

doi:10.1090/conm/369/06807

Cited by 28 publications

(57 citation statements)

References 2 publications

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“…For instance, we show in Section 6.3 how to classify C + -actions or, equivalently, A 1 -fibrations on Gizatullin surfaces; see Theorem 6.3.18. In Section 6.4 we strengthen our previous uniqueness result [FlZa3] for A 1 -fibrations on (singular, in general) Gizatullin surfaces.…”

Section: 02supporting

confidence: 79%

“…In [FlZa3,FKZ3] we have shown that a C * -action on a smooth affine surface V is unique up to equivalence unless V belongs to one of the classes (1)-(3) in Theorem 1.0.1 (see also 3.5.2 and 3.5.3 below). A similar uniqueness theorem holds for A 1 -fibrations V → A 1 ; see [FKZ3,5.13] and Proposition 6.4.1 in Section 6.4. Thus Theorem 1.0.1 is a consequence of the following result.…”

Section: 02mentioning

confidence: 98%

“…For symmetric reconstructions the following hold (see Proposition 4.6 in [FKZ3] The following proposition follows from the connectedness part of Theorem I.1.2 in [DaGi]. We provide an independent proof relying on [FKZ1].…”

Section: Preliminariesmentioning

confidence: 98%

“…The reversion of a zigzag, regarded as a birational transformation of the weighted dual graph, admits a factorization into a sequence of inner elementary transformations (see 1.4 in [FKZ3] and [FKZ1]). By an inner elementary transformation of a weighted graph we mean blowing up at an edge incident to a 0-vertex of degree 2 and blowing down the image of this vertex.…”

Section: Preliminariesmentioning

confidence: 99%

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Smooth affine surfaces with non-unique ℂ*-actions

2010

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In this paper we complete the classification of effective C * -actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion λ → λ −1 of C * . If a smooth affine surface V admits more than one C * -action, then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In [Transformation Groups 13:2, 2008, pp. 305-354] we gave a sufficient condition, in terms of the Dolgachev-Pinkham-Demazure (or DPD) presentation, for the uniqueness of a C * -action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V which is neither toric nor Danilov-Gizatullin, then V admits a continuous family of pairwise non-conjugated C * -actions depending on one or two parameters. We give an explicit description of all such surfaces and their C * -actions in terms of DPD presentations. We also show that for every k > 0 one can find a Danilov-Gizatullin surface V (n) of index n = n(k) with a family of pairwise non-conjugate C + -actions depending on k parameters. Contents

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Section: 02supporting

confidence: 79%

Section: 02mentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

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Smooth affine surfaces with non-unique ℂ*-actions

2010

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“…We get the following corollary as an immediate consequence, see also [9]. For the classification of Z d -homogeneous fibration with C * fibers we first state the non-equivariant version used in [6], see also [23].…”

Section: See That a Is A D-th Root Of Unity And Thus (3) Implies (1)mentioning

confidence: 93%

The algebraic density property for affine toric varieties

Kutzschebauch

Leuenberger

Liendo

2015

Journal of Pure and Applied Algebra

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In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Anders\'en-Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X-Y is different from T. For toric surfaces we are able to classify those which posses a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.Comment: 15 pages. Minor corrections. To appear in Journal of Pure and Applied Algebr

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The automorphism group of a rigid affine variety

Arzhantsev

Sergey

2016

Math. Nachr.

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An irreducible algebraic variety $X$ is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group $\text{Aut}(X)$ of a rigid affine variety contains a unique maximal torus $\mathbb{T}$. If the grading on the algebra of regular functions $\mathbb{K}[X]$ defined by the action of $\mathbb{T}$ is pointed, the group $\text{Aut}(X)$ is a finite extension of $\mathbb{T}$. As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.Comment: 12 page

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On the uniqueness of ℂ*-actions on affine surfaces

Cited by 28 publications

References 2 publications

Smooth affine surfaces with non-unique ℂ*-actions

Smooth affine surfaces with non-unique ℂ*-actions

The algebraic density property for affine toric varieties

The automorphism group of a rigid affine variety

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