by  S.Yu. Orevkov

Abstract:

The fundamental group of   C² - K   is computed where   K   is an algebraic curve having only simple double points (nodes) and satisfying a certain restriction at infinity. This restriction called the negativity condition at the infinity   topologically means that the braid at the infinity is positive (not necessarily strictly positive as it is claimed in the review in MathSciNet). This condition is satisfied, for example, for a nodal curve in   CP²,  for a generic curve in C² paramertized by two polynomials of given degrees, and also for a generic curve with a given Newton polygon. As corollary, a new proof of the Fulton-Deligne theorem (Zariski's conjecture) is obtained. It states that the fundamental group of   CP² - C   is abelian if   C   is a nodal curve, i.e. a curve which has only simple double points (nodes) as singularities.