On the existence of Ulrich bundles on geometrically ruled surfaces

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 $e\ge 0$, slightly extending a recent result due to M. Aprodu, L. Costa and R.M. Miró-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 $p$.

This paper has been withdrawn by the author due to a mistake in the proof of Theorem 1.3 which is not true in the form indicated in the paper. The paper will be resubmitted after a complete revision

References in corpus (1)

• An introduction to Ulrich bundles