Characterization of Ulrich bundles on Hirzebruch surfaces

Abstract

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. 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 $(\text{rk}(E),c_1(E))$ and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.

23 pages. Incorporated Section 5 in the other sections. Added Section 6 on existence and moduli space of Ulrich bundles. Final version in Revista Matematica Complutense