Higher-order Alexander invariants of plane algebraic curves

Abstract

We define new higher-order Alexander modules An(C) and higher-order degrees δn(C) which are invariants of the algebraic planar curve C. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve C. These invariants are in the spirit of those developed by Cochran and Harvey, which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.