Degenerations of scrolls to unions of planes

We study degenerations of scrolls to union of planes, a problem already considered by G. Zappa in [23] and [24]. We prove, using techniques different from those of Zappa, a result on degeneration to a union of planes with the mildest possible singularities, for linearly normal scrolls of genus g and of degree d >= 2g + 4 in Pd-2g+1. We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally, we review Zappa's original approach.