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An introduction to Ulrich bundles

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Abstract

After recalling the definition and basic properties of Ulrich bundles, we focus on the existence problem: does every smooth projective variety carry an Ulrich bundle? We show that the Serre construction provides a positive answer on certain surfaces and threefolds.

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Notes

  1. The case of abelian surfaces was treated in [6], and that of Enriques surfaces in [7] and [9].

References

  1. 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. arXiv:1212.6248

  2. Arrondo, E.: A home-made Hartshorne–Serre correspondence. Rev. Mat. Complut. 20(2), 423–443 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arrondo, E., Costa, L.: Vector bundles on Fano \(3\)-folds without intermediate cohomology. Comm. Algebra 28(8), 3899–3911 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beauville, A.: Determinantal hypersurfaces. Michigan Math. J. 48, 39–64 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beauville, A.: Vector bundles on the cubic threefold. In: Bertram, A., Carlson, J.A., Kley, H. (eds.) Symposium in Honor of C.H. Clemens. Contemporary Mathematics, vol. 312, pp. 71–86. American Mathematical Society, Providence (2002)

    Chapter  Google Scholar 

  6. Beauville, A.: Ulrich bundles on abelian surfaces. Proc. Amer. Math. Soc. 144(11), 4609–4611 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beauville, A.: Ulrich bundles on surfaces with \(q=p_g=0\) (2016). arXiv:1607.00895

  8. Borisov, L., Nuer, H.: Ulrich bundles on Enriques surfaces (2016). arXiv:1606.01459

  9. Casnati, G.: Special Ulrich bundles on non-special surfaces with \(p_g=q=0\) (2016). arXiv:1609.07915

  10. Druel, S.: Espace des modules des faisceaux de rang \(2\) semi-stables de classes de Chern \(c_1=0\), \(c_2=2\) et \(c_3=0\) sur la cubique de \(\mathbb{P}^4\). Int. Math. Res. Not. 2000(19), 985–1004 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eisenbud, D., Schreyer, F.-O., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16(3), 537–579 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hartshorne, R., Hirschowitz, A.: Smoothing algebraic space curves. In: Casas-Alvero, E., Welters, G., Xambó-Descamps, S. (eds.) Algebraic Geometry Sitges. Lecture Notes in Mathematics, vol. 1124, pp. 98–131. Springer, Berlin (1985)

    Google Scholar 

  13. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  14. Herzog, J., Ulrich, B., Backelin, J.: Linear maximal Cohen–Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71(2–3), 187–202 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Iskovskikh, V.A., Prokhorov, Yu.G.: Fano varieties. In: Parshin, A.N., Shafarevich, I.R. (eds.) Algebraic Geometry V. Encyclopaedia of Mathematical Sciences, vol. 47. Springer, Berlin (1999)

  16. Kapranov, M.M.: On the derived categories of coherent sheaves on some homogeneous spaces. Invent. Math. 92(3), 479–508 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mumford, D.: Lectures on Curves on an Algebraic Surface. Annals of Mathematics Studies, vol. 59. Princeton University Press, Princeton (1966)

    MATH  Google Scholar 

  18. Markushevich, D., Tikhomirov, A.S.: The Abel–Jacobi map of a moduli component of vector bundles on the cubic threefold. J. Algebraic Geom. 10(1), 37–62 (2001)

    MathSciNet  MATH  Google Scholar 

  19. O’Grady, K.G.: Desingularized moduli spaces of sheaves on a \(K3 \). J. Reine Angew. Math. 512, 49–117 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Schröter, H.E.: Nachweis der 27 Geraden auf der allgemeinen Oberfläche dritter Ordnung. J. Reine Angew. Math. 62, 265–280 (1863)

    Article  MathSciNet  Google Scholar 

  21. Ulrich, B.: Gorenstein rings and modules with high numbers of generators. Math. Z. 188(1), 23–32 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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  1. CNRS — Laboratoire J.-A. Dieudonné, Université Côte d’Azur, Parc Valrose, 06108, Nice Cedex 2, France

    Arnaud Beauville

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  1. Arnaud Beauville
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Beauville, A. An introduction to Ulrich bundles. European Journal of Mathematics 4, 26–36 (2018). https://doi.org/10.1007/s40879-017-0154-4

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