Hilbert scheme of some threefold scrolls over the Hirzebruch surface F_1

TL;DR

This paper investigates the Hilbert schemes of certain threefold scrolls over the Hirzebruch surface F_1, demonstrating their smoothness and describing the general members of the relevant component.

Contribution

It establishes the irreducibility and generic smoothness of a component of the Hilbert scheme for these threefold scrolls and characterizes the general elements.

Findings

The Hilbert scheme component is irreducible.

The component is generically smooth.

The general threefold scroll is explicitly described.

Abstract

Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface 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.