An arc in A^2 is a pair of power series x(t), y(t) in a single variable t. Assume that it parametrizes a plane algebraic curve X locally at a point a. If X is smooth at a, the ratio s(t) = x'(t)/y'(t) describes the slope of the tangents of X nearby a. It is invariant under reparametrization of the arc and has an expression in the partial derivatives of the defining equation of X. The Nash modification of X is then defined as the Zariski-closure in A^2 x P^1 of the image of the triple x(t), y(t), s(t) (for generic t).

In the talk, we ask for *all* rational functions in the (higher) derivatives of x(t) and y(t) which are invariant under
reparametrization. These ratios will be called the "algebraic curvatures" of the curve at smooth points. A complete classification of these curvatures can be given, and implicit expressions in terms of the defining equation of X are shown to exist. This is then used to produce one-step resolutions of the curve, using the associated higher
Nash-modification (joint work with Hana Melanova). The goal of these investigations is to extend them later to
the case of surfaces.