Simple birational extensions of the polynomial ring \(\mathbb{C}^{[3]}\) Braid group techniques in complex geometry. IV: Braid monodromy of the branch curve \(S_ 3\) of \(V_ 3\to\mathbb{C}\mathbb{P}^ 2\) and application to \(\pi_ 1(\mathbb{C}\mathbb{P}^ 2-S_ 3,\ast)\) 