A braid monodromy presentation for the pure braid group

Abstract: In the present paper, we consider a ``lexicographic section'' of the braid arrangement, and give a presentation of the fundamental group of its complement using the braid monodromy technique. We show that the resulting presentation coincides with the modified Artin presentation given by Margalit--McCammond. Moreover, we generalize the construction to the Manin--Schechtman arrangement $MS(n,k)$, a generalization of the braid arrangement. In particular, we give a presentation of $\pi_1(\mathbb{C}⁵ \setminus MS(5,2))$, which is the simplest and nontrivial example of the Manin--Schechtman arrangements.