Higher degree Galois covers of ℂℙ1 ×T

Amram, Meirav; Goldberg, David

doi:10.2140/agt.2004.4.841

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Higher degree Galois covers of $\mathbb{CP}^1 \times T$

Meirav Amram and David Goldberg

Algebraic & Geometric Topology 4 (2004) 841–859

DOI: 10.2140/agt.2004.4.841

arXiv: math.AG/0410554

Abstract

Let $T$ be a complex torus, and $X$ the surface ${ℂ ℙ}^{1} \times T$ . If $T$ is embedded in ${ℂ ℙ}^{n - 1}$ then $X$ may be embedded in ${ℂ ℙ}^{2 n - 1}$ . Let $X_{Gal}$ be its Galois cover with respect to a generic projection to ${ℂ ℙ}^{2}$ . In this paper we compute the fundamental group of $X_{Gal}$ , using the degeneration and regeneration techniques, the Moishezon–Teicher braid monodromy algorithm and group calculations. We show that $π_{1} (X_{Gal}) = ℤ^{4 n - 2}$ .

Keywords

Galois cover, fundamental group, generic projection, Sieberg–Witten invariants

Mathematical Subject Classification 2000

Primary: 14Q10

Secondary: 14J80, 32Q55

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Publication

Received: 17 June 2004

Accepted: 6 October 2004

Published: 7 October 2004

Authors

Meirav Amram
Einstein Institute for Mathematics The Hebrew University Jerusalem Israel

David Goldberg
Mathematics Department Colorado State University Fort Collins CO 80523 USA

Volume 4, issue 2 (2004)