Irreducible components of Hilbert schemes of rational curves with given normal bundle

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a reducible Hilbert scheme parametrizing all the rational curves in Ps with a given decomposition type of the normal bundle. We also characterize smooth non- degenerate rational curves contained in rational normal scrolls in terms of the splitting type of their restricted tangent bundles and compute their normal bundles.