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

Abstract

Let $T$ be a complex torus, and $X$ the surface ${ℂ\mathbb{P}}^1\times T$. If $T$ is embedded in ${ℂ\mathbb{P}}^{n-1}$ then $X$ may be embedded in ${ℂ\mathbb{P}}^{2n-1}$. Let $X_{Gal}$ be its Galois cover with respect to a generic projection to ${ℂ\mathbb{P}}^2$. 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 ${\pi }_1\left(X_{Gal}\right)={\mathbb{Z}}^{4n-2}$.