Zariski's conjecture and related problems

Nori, Madhav

doi:10.24033/asens.1450

Cited by 162 publications

(114 citation statements)

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“…Only in the case of curves of positive self-intersection (with some additional hypotheses) have we found results in the literature, cf. [1], [19].…”

Section: Remark If One Uses the Classes Fi(a) Instead Of 2/i{a) As mentioning

confidence: 99%

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Kotschick

Matić

1995

Math. Proc. Camb. Phil. Soc.

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IntroductionOne of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formula There are a number of results on this question in the literature. They can be divided into two classes, those proved by classical topological methods, which apply to topologically locally flat embeddings, including the G-signature theorem, and those proved by methods of gauge theory, which apply to smooth embeddings only.On the classical side, there is the result of Kervaire and Milnor[10], based on Rokhlin's theorem, which shows that certain homology classes are not represented by spheres. A major step forward was made by Rokhlin[20] and Hsiang and Szczarba [8] who introduced branched covers to study this problem for divisible homology classes. If the integral homology class represented by an embedded surface £ c: X is divisible by k, then the corresponding cover of X of order k branched along £ is again a smooth 4-manifold, so there is an obvious inequality, simply from the existence of the covering, asserting that its second Betti number is at least the absolute value of its signature. In the case k = 2, this gives

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“…Only in the case of curves of positive self-intersection (with some additional hypotheses) have we found results in the literature, cf. [1], [19].…”

Section: Remark If One Uses the Classes Fi(a) Instead Of 2/i{a) As mentioning

confidence: 99%

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Kotschick

Matić

1995

Math. Proc. Camb. Phil. Soc.

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“…Combining our main result with Nori's lemma [15] (see Proposition 3.1), we obtain the following: As the next application, we investigate the fundamental group of the complement of the Grassmannian dual variety, and prove a hyperplane section theorem of Zariski-Lefschetz-van Kampen type.…”

Section: Introductionmentioning

confidence: 79%

Generalized Zariski–van Kampen Theorem and Its Application to Grassmannian Dual Varieties

Shimada

2010

Int. J. Math.

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“…If d > 6a+ 2b, then by a result of Nori (cf. [19], but see also [13]), it follows that π 1 (CP 2 −C) is abelian. If we choose a generic line H 'at infinity' and set C =C − H, then as in 3.5 it follows that π 1 (C 2 − C) is also abelian.…”

Section: Examplesmentioning

confidence: 89%

Higher-order Alexander invariants of plane algebraic curves

Leidy¹,

Maxim²

2006

International Mathematics Research Notices

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Abstract. We define new higher-order Alexander modules A n (C) and higher-order degrees δ n (C) which are invariants of the algebraic planar curve C. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve C. These invariants are in the spirit of those developed by T. Cochran in [1] and S. Harvey in [7] and [8], which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higherorder degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.

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Zariski's conjecture and related problems

Cited by 162 publications

References 0 publications

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Generalized Zariski–van Kampen Theorem and Its Application to Grassmannian Dual Varieties

Higher-order Alexander invariants of plane algebraic curves

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