Affine Lines on Affine Surfaces and the Makar–Limanov Invariant

Gurjar, R. V.; Masuda, Kayo; Miyanishi, Masayoshi; Russell, Peter B.

doi:10.4153/cjm-2008-005-8

Cited by 20 publications

(23 citation statements)

References 16 publications

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“…(1) and (2): Let C 2 be as above the image of a general fiber C 1 of p 1 . By the (GSP), we know that C 2 is a smooth complete curve on V 2 , whence p a (C 2 ) = h. It follows from Theo-…”

Section: Lemma 12mentioning

confidence: 99%

“…(3) Note thatX is an ML 1 -surface, or equivalently a smooth affine surface with a unique A 1 -fibration parametrized by an affine curve, if and only if so is X (see [1]). HenceX 2 has a unique A 1 -fibration over an affine curve.…”

Section: Proofmentioning

confidence: 99%

“…Let X be an affine pseudo-plane as defined in the introduction (see also [1,7]). We fix an A 1 -fibration ρ : X → B, where B ∼ = A 1 .…”

Section: Case Of Affine Pseudo-planesmentioning

confidence: 99%

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A geometric approach to the Jacobian conjecture in dimension two

Miyanishi

2006

Journal of Algebra

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“…(1) and (2): Let C 2 be as above the image of a general fiber C 1 of p 1 . By the (GSP), we know that C 2 is a smooth complete curve on V 2 , whence p a (C 2 ) = h. It follows from Theo-…”

Section: Lemma 12mentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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A geometric approach to the Jacobian conjecture in dimension two

Miyanishi

2006

Journal of Algebra

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“…Professor M. Miyanishi and the referees informed us that Theorem 1.1 and Corollary 1.2 were proved jointly by Gurjar, Masuda, Miyanishi and Russell [12].…”

Section: Introductionmentioning

confidence: 99%

Affine lines on ℚ-homology planes with logarithmic Kodaira dimension -∞

Kishimoto

Kojima

2006

Transformation Groups

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In the present paper, we study a topologically contractible irreducible algebraic curve C on a Q-homology plane S with κ(S) = −∞. We determine such a pair (S, C) when κ(S\C) 0 and C is smooth. Moreover, we prove that if C is not smooth, then C has exactly one singular point and the Makar-Limanov invariant of S is trivial. IntroductionThroughout the present paper, we work over the complex number field C. A Qhomology plane is, by definition, a smooth algebraic surface S such that H i (S, Q) = (0) for any positive integer i. Similarly, a Z-homology plane is defined. It is well known that any Q-homology plane is affine and rational (cf.[7], [17]).Let C ⊂ C 2 be a topologically contractible irreducible algebraic curve. The Abhyankar-Moh-Suzuki theorem (cf.[1] and [33]) (resp., the Lin-Zaidenberg theorem (cf.[26])) says that if C is smooth (resp., C is not smooth), then there exists a system of coordinates {X, Y } on C 2 such that the curve C is defined by {X = 0} (resp., {X m = Y n }, where 0 < m < n and gcd(m, n) = 1). In particular, if C is smooth (resp., C is not smooth), then κ(C 2 \C) = −∞ (resp., κ(C 2 \C) = 1 and C has exactly one singular point). Later on, several authors studied topologically contractible algebraic curves on Q-homology planes. Zaidenberg [35] proved that a Z-homology plane of logarithmic Kodaira dimension 2 (resp., a Z-homology plane ∼ = C 2 of logarithmic Kodaira dimension 1) contains no topologically contractible algebraic curves (resp., a unique topologically contractible algebraic curve; it is actually smooth). Miyanishi and Tsunoda [31], Gurjar and Miyanishi [14] and Gurjar and Parameswaran [18] studied topologically contractible algebraic curves on Q-homology planes with nonnegative logarithmic Kodaira dimension. More precisely, the following results are known.Theorem 0.1. Let S be a Q-homology plane with κ(S) 0. Then the following assertions hold:(1) (cf.[31], [14]). If κ(S) = 2, then S contains no topologically contractible algebraic curves. TAKASHI KISHIMOTO, AND HIDEO KOJIMA(2) (cf. [14]). If κ(S) = 1, then S contains at least one and at most two topologically contractible algebraic curves. Moreover, such contractible curves are smooth. (3) (cf. [18]). If κ(S) = 0, then S contains at most two topologically contractible algebraic curves. Moreover, such contractible curves are smooth. (In fact, Gurjar and Parameswaran obtained more precise results. See [18] for more details.)By Theorem 0.1 we know that a Q-homology plane has logarithmic Kodaira dimension −∞ if and only if it contains at least three contractible algebraic curves. Note that, by the structure theorem of smooth affine surfaces with κ = −∞ (cf. Lemma 2.2), any smooth affine surface with κ = −∞ contains infinitely many affine lines.In the present paper, we shall study topologically contractible algebraic curves on Q-homology planes with κ = −∞ and attempt to give the results concerning more corollaries of the Abhyankar-Moh-Suzuki theorem and the Lin-Zaidenberg theorem. The main results (Theorems 1.1 and 1.3 and Corollary 1...

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“…In fact, if r = 1, then L as well as the boundary divisor ∞ − S − L is a linear chain. Hence X is an ML 0 -surface[7]. But the following argument works in the case r = 1 as well, provided the Gm-action stabilizes the component A since H 1 is a branching component of L + A.…”

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confidence: 99%

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 Lines on Affine Surfaces and the Makar–Limanov Invariant

Cited by 20 publications

References 16 publications

A geometric approach to the Jacobian conjecture in dimension two

A geometric approach to the Jacobian conjecture in dimension two

Affine lines on ℚ-homology planes with logarithmic Kodaira dimension -∞

Affine pseudo-planes with torus actions

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