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arXiv · 1806.10380

Characterization of Ulrich bundles on Hirzebruch surfaces

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

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Vincenzo Antonelli. 2020-03-11. Characterization of Ulrich bundles on Hirzebruch surfaces. https://doi.org/10.1007/s13163-019-00346-7

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