Hilbert scheme of some threefold scrolls over the Hirzebruch surface ${\mathbb F}_1$

Abstract

Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface $\mathbb{F}_1$ are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.