Affine lines on logarithmic Q-homology planes

Gurjar, R. V.; Miyanishi, Masayoshi

doi:10.1007/bf01934336

Cited by 20 publications

(14 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To compare Example 2.1 and Example 2.2, note that there exists no smooth irreducible affine surface that is contractible and contains two nonequivalent copies of C. Indeed, smooth homology planes of logarithmic Kodaira dimension one or two, contain at most one copy of C and smooth homology planes of logarithmic Kodaira dimension zero do not exist; see e.g. [GM92]. If the logarithmic Kodaira dimension of a smooth, contractible affine surface is −∞, then it must be C 2 by Miyanishi's characterization of the affine plane; see [Miy75] and [Miy84].…”

Section: Examples Of Varieties That Contain Non-equivalent Embeddingsmentioning

confidence: 99%

Uniqueness of embeddings of the affine line into algebraic groups

Feller

Santen

2019

J. Algebraic Geom.

View full text Add to dashboard Cite

show abstract

Section: Examples Of Varieties That Contain Non-equivalent Embeddingsmentioning

confidence: 99%

Uniqueness of embeddings of the affine line into algebraic groups

Feller

Santen

2019

J. Algebraic Geom.

View full text Add to dashboard Cite

show abstract

“…S 0 \L contains no complete curve. It is shown in [GM2] that S 0 \L contains an open U which is relatively minimal such that χ(U ) χ(S 0 \L). Moreover, if U 0 is a strictly minimal model of U , then χ(U 0 ) χ(U ).…”

Section: Propositionmentioning

confidence: 99%

Contractible affine surfaces with quotient singularities

Koras

Russell

2007

Transformation Groups

View full text Add to dashboard Cite

We consider contractible affine surfaces of negative Kodaira dimension with only quotient singularities. We prove that the smooth locus of such a surface has negative Kodaira dimension. It follows that if such a surface has only one singular point, then it is isomorphic to a quotient C 2 /G, where G is a finite group acting linearly on C 2 . The main resultIn [KR] we studied surfaces of the form C 3 / /C * and showed that they are isomorphic to C 2 /G, G a finite cyclic group acting linearly on C 2 . (If H is a reductive algebraic group acting algebraically on the affine variety X = Spec A, we denote by X/ /H the categorical quotient Y = Spec A H with A H the ring of invariants in A. If H is finite, we can identify X/ /H with X/H, the space of H-orbits.) Our results in [KR] came close to a characterization of C 2 /G via Kodaira dimension in case G is cyclic. Here we take up this problem for general finite subgroups of GL 2 (C). Note that we can assume that G is "small", i.e., does not contain any pseudoreflections. We prove the following main result.1.1. Theorem. Let S be a normal, topologically contractible affine surface over C of negative Kodaira dimension and smooth except for quotient singular points. Then:(i) the smooth locus S 0 of S has negative Kodaira dimension; and (ii) if S has at most one singular point, then S C 2 /G where G is a finite group acting linearly on C 2 .A complete description of S in the case of several singular points is given in Section 3. We note that by the Kodaira dimension of a surface X we mean, by definition, the Kodaira dimension of a desingularization X of X (see Paragraph 2.4.2 for more details).If S is smooth, then G = {1} and the fact that S is isomorphic to C 2 is a fundamental result of Miyanishi and Sugie [MS1] and Fujita [Fu1]. Other characterizations of C 2 /G, not involving Kodaira dimension, have been given. Miyanishi [M2] has shown that an 294 MARIUSZ KORAS AND PETER RUSSELL affine surface S dominated by C 2 via a finite morphism is isomorphic to C 2 /G, and Gurjar and Shastri [GS] proved the same if S is contractible and its fundamental group at infinity is finite. (In case S is smooth, this is a celebrated result of Ramanujam [R].) It appears, however, that these conditions are very difficult to verify in applications, even in situations where our condition on the Kodaira dimension is easily checked. As an example we discuss some consequences of our result for surfaces of the form C n / /H in Section 8 (see Theorem 8.1).The proof of Theorem 1.1 occupies Sections 2 to 7. Let κ be the Kodaira dimension of S 0 . We rule out κ = 0, 1 in Section 4, and κ = 2, as usual the most difficult case, in Sections 5, 6, and 7. In basic outline, our arguments follow those of [K] for κ = 0, 1 and of [KR] for κ = 2, and we do not hesitate to use results from those papers where they are directly applicable to our present more general situation. In case κ < 0, the proof of Theorem 1.1(ii), given in Section 3, is a quite direct consequence of the important result of Miyanishi and Tsu...

show abstract

“…[14]). If κ(S) = 1, then S contains at least one and at most two topologically contractible algebraic curves.…”

Section: Takashi Kishimoto and Hideo Kojimamentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Kishimoto

Kojima

2006

Transformation Groups

View full text Add to dashboard Cite

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...

show abstract

Affine lines on logarithmic Q-homology planes

Cited by 20 publications

References 10 publications

Uniqueness of embeddings of the affine line into algebraic groups

Uniqueness of embeddings of the affine line into algebraic groups

Contractible affine surfaces with quotient singularities

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

Contact Info

Product

Resources

About