Affine pseudo-planes and cancellation problem

Masuda, Kayo; Miyanishi, Masayoshi

doi:10.1090/s0002-9947-05-04046-8

Cited by 11 publications

(14 citation statements)

References 18 publications

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“…On the other hand, it follows from Theorem 13 that the cylinders S n,2 × A 1 , n ≥ 2, are all isomorphic. We thus recover a particular case of non-cancellation for so-called affine pseudo-planes studied in [22].…”

Section: Sketch Of Proofmentioning

confidence: 63%

$\mathbb{A}^1$-cylinders over smooth $\mathbb{A}^1$-fibered surfaces

Dubouloz

2020

Preprint

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We give a general structure theorem for affine A 1 -fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1 -fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve fails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.

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Section: Sketch Of Proofmentioning

confidence: 63%

$\mathbb{A}^1$-cylinders over smooth $\mathbb{A}^1$-fibered surfaces

Dubouloz

2020

Preprint

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“…This is the case for every affine pseudo-plane q : X → Z of type (m, r) with an integer r ≥ 1 such that r ≡ 1 (mod m). For the definition of an affine pseudo-plane of type (m, r), see [31] where the type is denoted by (d, r) instead of (m, r). In particular, q : X → Z is an A 1 -fibration with a unique singular fiber F 0 of multiplicity m > 1, i.e., F 0 = mA 1 and Z ∼ = A 1 .…”

Section: Homology Threefolds With a 1 -Fibrationsmentioning

confidence: 99%

“…Let H(m) = Z/mZ be the covering group which is identified with the m-th roots of unity. By [31,Lemma 2.6], X(m, r) is isomorphic to a hypersurface in A 3 = Spec C[x, y, z] defined by…”

Section: Homology Threefolds With a 1 -Fibrationsmentioning

confidence: 99%

$\mathbb{A}^{1}_{\ast}$-fibrations on affine threefolds

Gurjar¹,

Koras²,

Masuda³

et al. 2013

Affine Algebraic Geometry

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The main theme of the present article is A 1 * -fibrations defined on affine threefolds. The difference between A 1 * -fibration and the quotient morphism by a G m -action is more essential than in the case of an A 1 -fibration and the quotient morphism by a G aaction. We consider necessary (and partly sufficient) conditions under which a given A 1 * -fibration becomes the quotient morphism by a G m -action. Then we consider flat A 1 * -fibrations which are expected to be surjective, but this turns out to be not the case by an example of Winkelmann [47]. This example gives also a quasifinite endomorphism of A 2 which is not surjective [14]. We consider then the structure of a smooth affine threefold which has a flat A 1fibration or a flat A 1 * -fibration. More precisely, we consider affine threefolds with the additional condition that they are contractible.

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“…The authors generalized tom Dieck's examples in [11]. Our proof of Theorem 1.1 provides supplementary information on geometry of the unique A 1 -fibration on the given affine pseudo-plane interacting with the A 1 * -fibration induced by a given G m -action.…”

Section: Introductionmentioning

confidence: 95%

Affine pseudo-planes with torus actions

Miyanishi

Masuda

2006

Transformation Groups

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An affine pseudo-plane X is a smooth affine surface defined over C which is endowed with an A 1 -fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic Gm-action on X and X is an ML 1 -surface, we shall show that the universal covering X is isomorphic to an affine hypersurface x r y = z d − 1 in the affine 3-space A 3 and X is the quotient of X by the cyclic group Z/dZ via the action (x, y, z) → (ζx, ζ −r y, ζ a z), where r 2, d 2, 0 < a < d and gcd(a, d) = 1. It is also shown that a Q-homology plane X with κ(X) = −∞ and a nontrivial Gm-action is an affine pseudo-plane. The automorphism group Aut(X) is determined in the last section.

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Affine pseudo-planes and cancellation problem

Cited by 11 publications

References 18 publications

$\mathbb{A}^1$-cylinders over smooth $\mathbb{A}^1$-fibered surfaces

$\mathbb{A}^1$-cylinders over smooth $\mathbb{A}^1$-fibered surfaces

$\mathbb{A}^{1}_{\ast}$-fibrations on affine threefolds

Affine pseudo-planes with torus actions

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