The fundamental group of the complement of the branch curve of the second Hirzebruch surface

In this paper we prove that the Hirzebruch surface F 2 , ( 2 , 2 ) embedded in C P 17 supports the conjecture on the structure and properties of fundamental groups of complement of branch curves of generic projections, as laid out in [M. Teicher, New Invariants for surfaces, Contemp. Math. 231 (1999) 271–281]. We use the regeneration from [M. Friedman, M. Teicher, The regeneration of a 5-point, Pure and Applied Mathematics Quarterly 4 (2) (2008) 383–425. Fedor Bogomolov special issue, part I], the van Kampen theorem and properties of B ̃ n -groups [M. Teicher, On the quotient of the braid group by commutators of transversal half-twists and its group actions, Topology Appl. 78 (1997) 153–186], where B ̃ n is a quotient of the braid group B n , for n = 16 .