Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 1307–1326 | Cite as

Vector bundles whose restriction to a linear section is Ulrich

  • Rajesh S. Kulkarni
  • Yusuf Mustopa
  • Ian Shipman


An Ulrich sheaf on an n-dimensional projective variety \(X \subseteq \mathbb {P}^{N}\) is an initialized ACM sheaf which has the maximum possible number of global sections. Using a construction based on the representation theory of Roby–Clifford algebras, we prove that every normal ACM variety admits a reflexive sheaf whose restriction to a general 1-dimensional linear section is Ulrich; we call such sheaves \(\delta \)-Ulrich. In the case \(n=2,\) where \(\delta \)-Ulrich sheaves satisfy the property that their direct image under a general finite linear projection to \(\mathbb {P}^2\) is a semistable instanton bundle on \(\mathbb {P}^{2}\), we show that some high Veronese embedding of X admits a \(\delta \)-Ulrich sheaf with a global section.



I.S. was partially supported during the preparation of this paper by National Science Foundation award DMS-1204733. R. K. was partially supported by the National Science Foundation awards DMS-1004306 and DMS-1305377. We would like to thank the referee for helpful comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Rajesh S. Kulkarni
    • 1
  • Yusuf Mustopa
    • 2
  • Ian Shipman
    • 3
  1. 1.Michigan State UniversityEast LansingUSA
  2. 2.Tufts UniversityMedfordUSA
  3. 3.Harvard UniversityCambridgeUSA

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