The fundamental group of the galois cover of Hirzebruch surface F 1 (2, 2)

Meirav Amram; Mina Teicher; Uzi Vishne

doi:10.1142/S0218196707003780

The fundamental group of the galois cover of Hirzebruch surface F ₁ (2, 2)

Meirav Amram, Mina Teicher, Uzi Vishne

Department of Mathematics

Hebrew University of Jerusalem

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

5 Scopus citations

Abstract

This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces F_k(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is ℤ_c^n-2 where c = gcd(a, b) and n = 2ab + kb². In this paper, we degenerate the Hirzebruch surface Fi(2, 2), compute the braid monodromy factorization of the branch curve in ℂ², and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F₁ (2, 2) with respect to a generic projection is isomorphic to ℤ₂¹⁰.

Original language	English
Pages (from-to)	507-525
Number of pages	19
Journal	International Journal of Algebra and Computation
Volume	17
Issue number	3
DOIs	https://doi.org/10.1142/S0218196707003780
State	Published - May 2007

Bibliographical note

Funding Information:
author is partially supported by the Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany, and an Israel Science Foundation grant #8008/02-3 (Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties”), and by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).

Funding

author is partially supported by the Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany, and an Israel Science Foundation grant #8008/02-3 (Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties”), and by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).

Funders	Funder number
Emmy Noether Research Institute for Mathematics
Minerva Foundation
Israel Science Foundation	8008/02-3

Keywords

Fundamental group of complement of branch curve
Galois cover
Hirzebruch surfaces

Access to Document

10.1142/S0218196707003780

Cite this

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abstract = "This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is ℤcn-2 where c = gcd(a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface Fi(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1 (2, 2) with respect to a generic projection is isomorphic to ℤ210.",

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author = "Meirav Amram and Mina Teicher and Uzi Vishne",

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N2 - This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is ℤcn-2 where c = gcd(a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface Fi(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1 (2, 2) with respect to a generic projection is isomorphic to ℤ210.

AB - This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is ℤcn-2 where c = gcd(a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface Fi(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1 (2, 2) with respect to a generic projection is isomorphic to ℤ210.

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