Topology of generic line arrangements

TL;DR

This paper proves that for generic complex line arrangements, their defining polynomials are topologically equivalent up to small deformation, extending the result to hyperplane arrangements in higher dimensions.

Contribution

It generalizes the topological equivalence of defining polynomials from line arrangements to higher-dimensional hyperplane arrangements.

Findings

Generic complex line arrangements have topologically equivalent defining polynomials.

The result extends to hyperplane arrangements in higher dimensions within equivalent families.

Defines a framework for understanding the topological classification of arrangements.

Abstract

Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial, the product of a_ix+b_iy+c_i, so that A = (f=0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, topologically equivalent. In higher dimension the related result is that within a family of equivalent hyperplane arrangements the defining polynomials are topologically equivalent.