Fundamental groups of the complements to reducible curves on smooth surfaces with restricted classes of components

Abstract

We classify equisingular families of pencils of curves on smooth simply connected projective surfaces whose dual varieties have degree at most 6, and which contain sufficiently large number of members which irreducible components are either hyperplane sections or irreducible components of hyperplane sections. As a consequence, we describe the fundamental groups of complements to curves on such surfaces whose irreducible components satisfy the above condition and whose fundamental groups admit an essential surjection onto a free group of rank greater than 6.