Characterization of Ulrich bundles on Hirzebruch surfaces.

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.