# Ulrich Bundles on some threefold scrolls over $\mathbb{F}_e$

## TL;DR

This paper studies the existence, moduli, and properties of Ulrich vector bundles on certain threefold scrolls over Hirzebruch surfaces, providing explicit descriptions, dimension counts, and showing these varieties are Ulrich wild.

## Contribution

It explicitly describes moduli components of Ulrich bundles on threefold scrolls over Hirzebruch surfaces, including stability, dimension, and smoothness, and establishes their Ulrich wildness.

## Key findings

- Moduli spaces of Ulrich bundles are explicitly described.
- Components are shown to be generically smooth and contain stable, indecomposable bundles.
- The Ulrich complexity of these threefolds is computed, proving they are Ulrich wild.

## Abstract

We investigate the existence of Ulrich vector bundles on suitable $3$-fold scrolls $X_e$ over Hirzebruch surfaces $\mathbb{F}_e$, for any integer $e \geqslant 0$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{F}_e$ that are uniform in the sense of Brosius and Aprodu--Brinzanescu. We explicitely describe components of moduli spaces of rank $r \geqslant 1$ vector bundles which are Ulrich with respect to the tautological polarization on $X_e$ and whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components, proving also that they are generically smooth. As a direct consequence of these facts, we also compute the Ulrich complexity of any such $X_e$ and give an effective proof of the fact that these $X_e$'s turn out to be geometrically Ulrich wild. At last, the machinery developed for $3$--fold scrolls $X_e$ allows us to deduce Ulrichness results on rank $r \geqslant 1$ vector bundles on $\mathbb{F}_e$, for any $e \geqslant 0$, with respect to a naturally associated (very ample) polarization.