In the paper under review, the authors investigate the fundamental group of the Galois cover of the \((2,3)\)-embedding of \(X:=\mathbb{P}^{1}_{\mathbb{C}} \times T\), where \(T\) is a complex torus in \(\mathbb{P}^{2}_{\mathbb{C}}\). More precisely, since \(\mathbb{P}^{1}_{\mathbb{C}}\) can be embedded into \(\mathbb{P}^{m}_{\mathbb{C}}\) with \(m\geq 2\) and \(T\) can be embedded into \(\mathbb{P}^{n-1}_{\mathbb{C}}\) with \(n\geq 3\), then \(\mathbb{P}^{1}_{\mathbb{C}} \times T\) can be embedded into a larger projective space by the Segre embedding of \(\mathbb{P}^{m}_{\mathbb{C}} \times \mathbb{P}^{n-1}_{\mathbb{C}}\). Such an embedding is the \((m,n)\)-embedding of \(\mathbb{P}^{1}_{\mathbb{C}} \times T\) and one denotes the image of the embedding by \(X_{m,n}\). The resulting surface is of general type with index zero. In order to investigate the fundamental group, the authors apply Moishezon-Teicher’s algorithm combined with certain degeneration techniques.