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.

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Acknowledgements

The author wants to thank G. Casnati and F. Malaspina for the helpful discussions on the subject. The author also thanks the anonymous referees for the useful suggestions and remarks which have improved the whole exposition.

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Correspondence to Vincenzo Antonelli.

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Antonelli, V. Characterization of Ulrich bundles on Hirzebruch surfaces. Rev Mat Complut 34, 43–74 (2021). https://doi.org/10.1007/s13163-019-00346-7

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Keywords

  • Vector bundles
  • Ulrich bundles
  • Ruled surfaces
  • Beilinson-type spectral sequence

Mathematics Subject Classification

  • 14J60
  • 14F05
  • 14J26