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 [23] and [24]. We prove, using techniques different 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 $d\ge 2g+4$ in ${\mathbb{P}}^{d-2g+1}$. We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally we review Zappa's original approach.