On the existence of Ulrich bundles on geometrically ruled surfaces

TL;DR

This paper investigates the existence and construction of Ulrich bundles on geometrically ruled surfaces, extending previous results and providing new existence proofs for various ranks and conditions.

Contribution

It extends prior work by establishing the existence of Ulrich bundles of rank 2 on ruled surfaces with non-negative invariant and proves universal existence on elliptic ruled surfaces.

Findings

Ulrich line bundles and rank 2 Ulrich bundles exist on certain ruled surfaces.

On elliptic ruled surfaces, Ulrich bundles of rank at most 2 always exist.

Families of non-isomorphic, indecomposable Ulrich bundles of arbitrary large dimension are constructed.

Abstract

Let $S$ be a geometrically ruled surface with invariant $e$ on a curve $C$. We deal with Ulrich line bundles and $\mu$-stable special Ulrich bundles of rank $2$ on $S$ when $e\ge0$, slightly extending a recent result due to M. Aprodu, L. Costa and R.M. Mir\'o-Roig. If $C$ is elliptic, we also prove that $S$ always supports Ulrich bundles of rank at most $2$, without any restriction on $e$. Finally, we show that in many cases $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$.