A note concerning Fox’s paper on Fenchel’s conjecture

Chau, T. C.

doi:10.1090/s0002-9939-1983-0702279-5

Cited by 8 publications

(6 citation statements)

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“…The following two facts account for the significance of our construction. This was the Fenchel's conjecture proved by Fox [3], the proof of which contains an error later fixed by Chau [2]. Note that further by Poincaré's lemma in group theory, there exists a normal subgroup of finite index without elliptic elements although we do not need this latter fact.…”

Section: Geodesic Cover Of Co-finite Groupsmentioning

confidence: 77%

Geodesic cover of Fuchsian groups

2024

Annales Mathématiques Blaise Pascal

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Section: Geodesic Cover Of Co-finite Groupsmentioning

confidence: 77%

Geodesic cover of Fuchsian groups

2024

Annales Mathématiques Blaise Pascal

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“…By the solution of Fenchel's Conjecture due to Nielsen-Bundgaard and Fox (see [2]), there exists a finite Galois ramified morphism τ : C → C such that τ is ramified precisely over the points φ(G i ) with ramification index m i .…”

Section: Minimal P 1 -Fibrations As Quotientmentioning

confidence: 99%

“…The image of the line x 1 = x 2 is a curve with equation z 2 = az 2 1 for some non-zero constant z. Similarly, the images of the other two lines have equations z…”

Section: Examplesmentioning

confidence: 99%

On the fundamental group of a smooth projective surface with a finite group of automorphisms

Gurjar¹,

Purnaprajna²

2018

J. Math. Soc. Japan

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In this article we prove new results on fundamental groups for some classes of fibered smooth projective algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of the surfaces under study upto finite index. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori's well-known question on fundamental groups and free abelianness of second homotopy groups for these surfaces. We also prove a theorem that bounds the multiplicity of the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms G in terms of the multiplicities of the induced fibration on X/G. If X/G is a P 1 -fibration, we show that the multiplicity actually divides |G|. This theorem on multiplicity, which is of independent interest, plays an important role in our theorems.

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“…Consider the restrictions of T f and O W (2) to any fiber of f . Both these restrictions are of degree 2.…”

Section: Proof Of Lemma 62mentioning

confidence: 99%

“…, m 3 be the multiplicities of the unique feather of each singular fiber. By the solution of Fenchel's Conjecture due to Nielsen-Bundagaard and Fox [5] (see also [2]), there exists a curve C and a Galois map g : C → C which is ramified precisely at f (F i ), 1 i n with ramification index m i . The normalized fiber product W × C C is again a P 1 -fibration such that each singular fiber has at least one reduced feather.…”

Section: Proof Of Lemma 62mentioning

confidence: 99%

On the Zariski-Lipman Conjecture for normal algebraic surfaces

Biswas

Gurjar

Kolte

2014

Journal of the London Mathematical Society

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We consider the Zariski-Lipman Conjecture on a free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several classes of affine and projective surfaces.THE ZARISKI-LIPMAN CONJECTURE 271As far as we can see, none of these results imply our results. Given a smooth algebraic surface V and a projective completionV of it such that D :=V − V is a divisor whose only singularities are nodes, the logarithmic Kodaira dimension of V , denoted byκ(V ), is defined as the supremum of the dimensions of the images ofV under the rational maps defined by H 0 (V , n(K + D)), n 1. If this linear system is trivial for all n 1, then we defineκ(V ) = −∞ (see [11]).Our approach to this conjecture in this paper is global and it uses the theory of non-complete algebraic surfaces developed by Iitaka, Kawamata, Fujita, Miyanishi, Sugie, Tsunoda and other Japanese mathematicians. This theory has proved to be very effective in the solution of many problems about non-complete algebraic surfaces, and the arguments in this paper are just one more instance of this. We also use some results from the theory of vector bundles on smooth projective curves. Although some of our arguments are valid assuming only projectivity of the module of derivations, for many arguments we need the full force of the assumption that the tangent bundle of the smooth locus is trivial. As can be seen from the somewhat involved proofs in this paper, this stronger hypothesis is justified by the difficulty of the general conjecture. In this paper we verify the conjecture for all affine surfaces V such thatκ(V − Sing(V )) 1, and prove the conjecture in almost all the cases when V is projective. In particular, we prove that if V is projective, the tangent bundle of V − Sing(V ) is trivial andκ(V − Sing(V )) = 0 or 1, then V is smooth.All the varieties in this paper will be assumed to be over an algebraically closed field k of characteristic 0. If V is an algebraic surface, then by V 0 we denote the smooth locus of V .We now state the results of this paper.Theorem 1.2. Let V be an algebraic surface defined over k. Assume that the tangent bundle of V 0 is trivial. The following statements hold.(1) If V is an affine algebraic surface such thatκ(V 0 ) 1, then V is smooth.(2) If V is a projective surface, thenκ(V 0 ) 0 and V has at most one singularity.(3) If V is a projective surface such thatκ(V 0 ) = 0, then V is smooth.(4) Assume that V is a projective surface such thatκ(V 0 ) = −∞. Let p be the unique singular point of V . Then there exists a resolution of singularities π : W → V such that there is a P 1 -fibration W → C, where C is a smooth projective curve. If the genus of C is at least 2, then V is smooth.(5) With the notation in (4), if C is a rational curve, then V is smooth.(6) With the notation in (4), let C be an elliptic curve. If at least one singular fiber of W → C has a non-reduced feather (defined later), then V is smooth. Remark 1.3. We ...

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A note concerning Fox’s paper on Fenchel’s conjecture

Cited by 8 publications

References 2 publications

Geodesic cover of Fuchsian groups

Geodesic cover of Fuchsian groups

On the fundamental group of a smooth projective surface with a finite group of automorphisms

On the Zariski-Lipman Conjecture for normal algebraic surfaces

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