Vanishing of Higher Order Alexander-type Invariants of Plane Curves

Abstract

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper we characterize the vanishing of such invariants for transversal unions of plane curves $C'$ and $C''$ in terms of the finiteness, the vanishing properties of the invariants of $C'$ and $C''$, and whether they are irreducible or not. As a consequence, we characterize which of these types of curves have trivial multivariable Alexander polynomial in terms of their defining equations. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.