Characterization of Ulrich bundles on Hirzebruch surfaces.

In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over 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. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs (rk (E) , c 1 (E)) and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks. [ABSTRACT FROM AUTHOR]