A note on some moduli spaces of Ulrich Bundles

TL;DR

This paper proves that certain moduli spaces of Ulrich bundles on 3-fold scrolls over Hirzebruch surfaces are generically smooth, irreducible, and unirational, providing new insights into their geometric structure.

Contribution

It establishes the generic smoothness, irreducibility, and unirationality of specific moduli spaces of Ulrich bundles on 3-fold scrolls over Hirzebruch surfaces, extending previous results.

Findings

Moduli spaces are generically smooth and unirational.

Stronger results show irreducibility and smoothness for associated moduli spaces.

Ulrich bundles are parametrized on suitable 3-fold scrolls over Hirzebruch surfaces.

Abstract

We prove that the modular component $\mathcal M(r)$, constructed in the Main Theorem of a former paper of us (published in Adv. Math on 2024), paramatrizing (isomorphism classes of) Ulrich vector bundles of rank $r$ and given Chern classes, on suitable $3$-fold scrolls $X_e$ over Hirzebruch surfaces $\mathbb{F}_{e\geq 0}$, which arise as tautological embeddings of projectivization of very-ample vector bundles on $\mathbb{F}_e$, is generically smooth and unirational. A stronger result holds for the suitable associated moduli space $\mathcal M_{\mathbb F_e}(r)$ of vector bundles of rank $r$ and given Chern classes on $\mathbb{F}_e$, Ulrich w.r.t. the very ample polarization $c_1({\mathcal E}_e) = \mathcal O_{\mathbb F_e}(3, b_e),$ which turns out to be generically smooth, irreducible and unirational.