Advertisement

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
Article
  • 84 Downloads

Abstract

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.

Notes

Acknowledgements

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.

References

  1. 1.
    Aprodu, M., Costa, L., Miro-Roig, R.: Ulrich bundles on ruled surfaces. Preprint arXiv:1609.08340
  2. 2.
    Aprodu, M., Farkas, G., Ortega, A.: Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces. J. Reine Angew. Math. doi: 10.1515/crelle-2014-0124
  3. 3.
    Beauville, A.: Ulrich bundles on abelian surfaces. Proc. Am. Math. Soc. 144(11), 4609–4611 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Backelin, J., Herzog, J., Sanders, H.: Matrix factorizations of homogeneous polynomials. In: Avramov, L., Tchakerian, K. (eds.) Algebra Some Current Trends (Varna, 1986), Volume 1352 of Lecture Notes in Mathematics, pp. 1–33. Springer, Berlin (1988)Google Scholar
  5. 5.
    Brennan, J.P., Herzog, J., Ulrich, B.: Maximally generated Cohen–Macaulay modules. Math. Scand. 61(2), 181–203 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Backelin, J., Herzog, J., Ulrich, B.: Linear maximal Cohen–Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71(2–3), 187–202 (1991)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Borisov, L., Nuer, H.: Ulrich bundles on Enriques surfaces. Preprint arXiv:1606.01459
  8. 8.
    Childs, L.N.: Linearizing of \(n\)-ic forms and generalized Clifford algebras. Linear Multilinear Algebra 5(4), 267–278 (1977/1978)Google Scholar
  9. 9.
    Coskun, I., Huizenga, J., Woolf, M.: Equivariant Ulrich Bundles on Flag Varieties. Preprint arXiv:1507.00102
  10. 10.
    Costa, L., Miró-Roig, R.M.: \(GL(V)\)-invariant Ulrich bundles on Grassmannians. Math. Ann. 361(1–2), 443–457 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Costa, L., Miró-Roig, R.M., Pons-Llopis, J.: The representation type of Segre varieties. Adv. Math. 230(4–6), 1995–2013 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Eisenbud, D., Schreyer, F.-O., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16(3), 537–579 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Faenzi, D., Pons-Llopis, J.: The CM representation type of projective varieties. Preprint arXiv:1504.03819
  14. 14.
    Hanes, D.: Special Conditions on Maximal Cohen–Macaulay Modules, and Applications to the Theory of Multiplicities. University of Michigan, Ph.D. thesis (1999)Google Scholar
  15. 15.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Jardim, M.: Instanton sheaves on complex projective spaces. Collect. Math. 57(1), 69–91 (2006)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kulkarni, R.S., Mustopa, Y., Shipman, I.: The characteristic polynomial of an algebra and representations. Preprint arXiv:1507.08361
  18. 18.
    Kulkarni, R.S., Mustopa, Y., Shipman, I.: Ulrich sheaves and higher-rank Brill–Noether theory. J. Algebra 474, 166–179 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Migliore, J., Nagel, U.: Survey article: a tour of the weak and strong Lefschetz properties. J. Commut. Algebra 5(3), 329–358 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Miró-Roig, R.M.: The representation type of rational normal scrolls. Rend. Circ. Mat. Palermo 62(1), 153–164 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (2011). Corrected reprint of the 1988 edition with an appendix by S. I. GelfandGoogle Scholar
  22. 22.
    Pappacena, C.J.: Matrix pencils and a generalized Clifford algebra. Linear Algebra Appl. 313(1–3), 1–20 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Procesi, C.: A formal inverse to the Cayley–Hamilton theorem. J. Algebra 107(1), 63–74 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Roby, N.: Algèbres de Clifford des formes polynomes. C. R. Acad. Sci. Paris Sér A–B 268, 484–486 (1969)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Van den Bergh, M.: Linearisations of binary and ternary forms. J. Algebra 109(1), 172–183 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

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

Personalised recommendations