Minimal stratifications for line arrangements and positive homogeneous presentations for fundamental groups

The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the "dual" objects so called minimal stratifications. The strata are explicitly described as semialgebraic sets. The stratification induces a partition of the complement into a disjoint union of contractible spaces, which is minimal in the sense that the number of codimension $k$ pieces equals the $k$-th Betti number. We also discuss presentations for the fundamental group associated to the minimal stratification. In particular, we show that the fundamental groups of complements of a real arrangements have positive homogeneous presentations.