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The fundamental group of a Galois cover of $$\mathbb{C} \mathbb{P}^1 \times T$$. (English) Zbl 1037.14006
Let $$T$$ be a complex torus in $$\mathbb{C}\mathbb{P}_2$$ and $$X= \mathbb{C}\mathbb{P}_1 \times T$$. By embedding $$X$$ into $$\mathbb{C}\mathbb{P}_5$$ by the usual Segre map $$\mathbb{C}\mathbb{P}_1 \times \mathbb{C}\mathbb{P}_2 \rightarrow \mathbb{C}\mathbb{P}_5$$ a generic projection $$f: X \rightarrow \mathbb{C}\mathbb{P}_2$$ is obtained by projecting $$X$$ from a general plane in $$\mathbb{C}\mathbb{P}_5 \setminus X$$ to $$\mathbb{C}\mathbb{P}_2$$. The Galois cover of $$X$$ is the closure of the $$n$$-fibred product $$X_{\text{gal}} = \overline{ X \times_{f} \ldots \times_{f} X - \Delta}$$ where $$n = \text{deg}(f)$$ and $$\Delta$$ is the generalized diagonal.
In this paper the authors compute $$\pi_1( X_{\text{gal}}) = \mathbb{Z}^{10}$$ stated as theorem 9.3, using braid monodromy techniques developed by B. Moishezon and M. Teicher [Invent. Math. 89, 601–643 (1987; Zbl 0627.14019)], the van Kampen theorem and various computational methods of groups.

##### MSC:
 14F35 Homotopy theory and fundamental groups in algebraic geometry 14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 12F10 Separable extensions, Galois theory 20F36 Braid groups; Artin groups 14E20 Coverings in algebraic geometry 14Q10 Computational aspects of algebraic surfaces 14N05 Projective techniques in algebraic geometry 14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
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