  We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and
higher-order degrees $\delta_n(C)$ 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 T. Cochran in \cite{C} and S. Harvey in
\cite{H} and \cite{Har}, 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.
