On automorphism groups of affine surfaces

Kovalenko, Sergei; Perepechko, Alexander; Zaidenberg, Mikhail

doi:10.2969/aspm/07510207

Cited by 15 publications

(18 citation statements)

References 52 publications

(162 reference statements)

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“…The notion of an ind-group goes back to Shafarevich who called these objects infinite dimensional groups (see [13]). We refer to [4] and [7,Section 2] for basic notions in this context. Definition 2.1.…”

Section: Preliminariesmentioning

confidence: 99%

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When is the automorphism group of an affine variety nested?

Perepechko¹,

Regeta²

2019

Preprint

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Section: Preliminariesmentioning

confidence: 99%

“…In [7] and [9] the neutral component Aut • (X) of the group of automorphisms Aut(X) of an affine surface X has been studied. Note that Aut • (X) is a closed subgroup of Aut(X).…”

Section: Introductionmentioning

confidence: 99%

When is the automorphism group of an affine variety nested?

Perepechko¹,

Regeta²

2019

Preprint

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“…Remark Using different techniques, it is proved in [, Proposition 4.4] that if the neutral component of the automorphism group of a rigid affine surface is an affine algebraic group, then this component is a torus. Let us notice that surfaces of class (

{ML}_{2}

) in terminology of are precisely rigid surfaces in our terms.…”

Section: Subtori In the Automorphism Groupmentioning

confidence: 99%

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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“…The next proposition can be deduced from Theorems 6.3 and 8.24 in [12] and their corollaries. However, in our particular case we prefer to give a simple direct argument.…”

Section: On Moduli Spaces Of Gdf Surfacesmentioning

confidence: 91%

“…The first equation in (9) gives (12). This means that U j+1 → U j is an isomorphism commuting with the projections to…”

Section: Gdf Surfaces Whose Fiber Trees Are Bushesmentioning

confidence: 99%

Cancellation for surfaces revisited. II

Flenner,

Kaliman,

Zaidenberg

2018

Preprint

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Let X and X ′ be affine algebraic varieties over a field k. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X×A n ≅ X ′ × A n implies X ≅ X ′ . In Part I of this paper (arXiv:1610.01805) we provided a criterion for cancellation in the case where X is a normal affine surface admitting an A 1 -fibration X → B over a smooth affine curve B. If X does not admit such an A 1 -fibration then the cancellation by the affine line is known to hold for X by a result of Bandman and Makar-Limanov. In the present Part II we classify all pairs (X, X ′ ) of smooth affine surfaces A 1 -fibered over B with only reduced fibers whose cylinders X × A 1 , X ′ × A 1 are isomorphic over B. Our criterion of isomorphism of cylinders over B is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of X over B. We construct a course moduli space of such surfaces with a given cylinder. This moduli space has an infinite number of irreducible components whose dimension grows. Each irreducible component of this moduli space is an affine variety with quotient singularities. Contents 18 6.2. Special isomorphisms and regularization 19 7. On moduli spaces of GDF surfaces 22 7.1. Coarse moduli spaces of GDF surfaces 23 7.2. The automorphism group of a GDF surface 24 7.3. Configuration spaces and configuration invariants 25 7.4. Versal deformation families of trivializing sequences 27 7.5. Proof of Theorem 7.3 31 References 33

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On automorphism groups of affine surfaces

Cited by 15 publications

References 52 publications

When is the automorphism group of an affine variety nested?

When is the automorphism group of an affine variety nested?

The automorphism group of a rigid affine variety

Cancellation for surfaces revisited. II

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