On some "sporadic" moduli spaces of Ulrich bundles on some 3-fold scrolls over $\mathbb{F}_0$

TL;DR

This paper explores the existence and structure of special Ulrich vector bundles on certain 3-fold scrolls over the Hirzebruch surface, revealing their deformation properties, moduli space components, and stability characteristics.

Contribution

It identifies and describes 'sporadic' Ulrich bundles and their moduli spaces on 3-fold scrolls over , including explicit descriptions of irreducible components and stability results.

Findings

Existence of sporadic Ulrich bundles on specific 3-fold scrolls.

Description of irreducible components of moduli spaces, some singleton, some positive-dimensional.

General points in moduli are slope-stable vector bundles.

Abstract

We investigate on the existence of some "sporadic", rank-$r \geqslant 1$ Ulrich vector bundles on suitable $3$-fold scrolls $X$ over the Hirzebruch surface $\mathbb{F}_0$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{F}_0$ that are uniform in the sense of Brosius and Aprodu--Brinzanescu. Such Ulrich bundles arise as deformations of ``iterative" extensions by means of "sporadic" Ulrich line bundles. We moreover explicitely describe irreducible components of the corresponding "sporadic" moduli spaces of rank $r \geqslant 1$ vector bundles which are Ulrich with respect to the tautological polarization on $X$. In some cases such irreducible components turn out to be a singleton, in some other cases such components are generically smooth, whose positive dimension has been computed and whose general point turns out to be a slope-stable vector bundle.