Volume 4, issue 2 (2004)

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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
 arXiv: math.AG/0410554
Abstract

Let $T$ be a complex torus, and $X$ the surface ${ℂℙ}^{1}×T$. If $T$ is embedded in ${ℂℙ}^{n-1}$ then $X$ may be embedded in ${ℂℙ}^{2n-1}$. Let ${X}_{Gal}$ be its Galois cover with respect to a generic projection to ${ℂℙ}^{2}$. 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 ${\pi }_{1}\left({X}_{Gal}\right)={ℤ}^{4n-2}$.

Keywords
Galois cover, fundamental group, generic projection, Sieberg–Witten invariants
Mathematical Subject Classification 2000
Primary: 14Q10
Secondary: 14J80, 32Q55