Degenerations of scrolls to unions of planes

Abstract. In this paper we study degenerations of scrolls to union of planes, a problem already considered by G. Zappa in [25] and [26]. We prove, using techniques dierent from the ones of Zappa, a degeneration result to union of planes with the mildest possible singularities, for linearly normal scrolls of genus g and of degree d 2g + 4 in P d2g+1. We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally we review Zappa's original approach. Dedicated to Professor G. Zappa on his 90th birthday 1