The Hilbert Scheme of degree two curves and certain ropes

We study families of ropes of any codimension that are supported on lines. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus are in the same component of the Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curvea have degree two. In this case the component is not reduced.