Hilbert schemes of some threefold scrolls over F_e

TL;DR

This paper investigates the structure and properties of Hilbert schemes for certain threefold scrolls over Hirzebruch surfaces, demonstrating smoothness and describing general elements, extending previous work for the case e=1.

Contribution

It extends the analysis of Hilbert schemes of threefold scrolls to cases where e > 1, showing irreducibility, smoothness, and describing the general points of the component.

Findings

Hilbert scheme component is generically smooth of expected dimension

General point of the component is explicitly described

The study generalizes previous results for e=1

Abstract

Hilbert schemes of suitable smooth, projective 3-fold scrolls over the Hirzebruch surface F_e, with e > 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. This article generalizes the study of Hilbert schemes done in arXiv:1110.5464 for e=1.