The fundamental group of the complement of the branch curve of ℂℙ1 × T

Meirav Amram; Michael Friedman; Mina Teicher

doi:10.1007/s10114-009-6473-8

The fundamental group of the complement of the branch curve of ℂℙ¹ × T

Meirav Amram^*, Michael Friedman, Mina Teicher

^*Corresponding author for this work

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Denoting by T the complex projective torus, we can embed the surface ℂℙ¹× T in ℂℙ⁵. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not "ample enough", the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable: a property which holds for these groups for "ample enough" embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group (as shown in this paper) give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.

Original language	English
Pages (from-to)	1443-1458
Number of pages	16
Journal	Acta Mathematica Sinica, English Series
Volume	25
Issue number	9
DOIs	https://doi.org/10.1007/s10114-009-6473-8
State	Published - Sep 2009
Externally published	Yes

Funding

Funders	Funder number
Deutscher Akademischer Austauschdienst	EU-network HPRN-CT-2009-00099
Minerva Foundation
Israel Science Foundation	8008/02-3

Keywords

Branch curve
Curves and singularities
Fundamental group
Generic projection

Access to Document

10.1007/s10114-009-6473-8

Cite this

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title = "The fundamental group of the complement of the branch curve of ℂℙ1 × T",

abstract = "Denoting by T the complex projective torus, we can embed the surface ℂℙ1× T in ℂℙ5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not {"}ample enough{"}, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable: a property which holds for these groups for {"}ample enough{"} embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group (as shown in this paper) give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.",

keywords = "Branch curve, Curves and singularities, Fundamental group, Generic projection",

author = "Meirav Amram and Michael Friedman and Mina Teicher",

note = "Funding Information: Received September 13, 2006, Accepted May 21, 2008 The third author is partially supported by DAAD and EU-network HPRN-CT-2009-00099(EAGER); The Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany; The Israel Science Foundation grant # 8008/02-3 (Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties”).",

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T1 - The fundamental group of the complement of the branch curve of ℂℙ1 × T

AU - Amram, Meirav

AU - Friedman, Michael

AU - Teicher, Mina

N1 - Funding Information: Received September 13, 2006, Accepted May 21, 2008 The third author is partially supported by DAAD and EU-network HPRN-CT-2009-00099(EAGER); The Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany; The Israel Science Foundation grant # 8008/02-3 (Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties”).

PY - 2009/9

Y1 - 2009/9

N2 - Denoting by T the complex projective torus, we can embed the surface ℂℙ1× T in ℂℙ5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not "ample enough", the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable: a property which holds for these groups for "ample enough" embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group (as shown in this paper) give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.

AB - Denoting by T the complex projective torus, we can embed the surface ℂℙ1× T in ℂℙ5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not "ample enough", the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable: a property which holds for these groups for "ample enough" embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group (as shown in this paper) give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.

KW - Branch curve

KW - Curves and singularities

KW - Fundamental group

KW - Generic projection

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