The fundamental group of a Galois cover of $\mathbb{CP}^1 \times T$

Abstract

Let $T$ be the complex projective torus, and $X$ the surface ${ℂ\mathbb{P}}^1\times T$. 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 ${\pi }_1\left(X_{Gal}\right)={\mathbb{Z}}^{10}$.