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Theta divisors and Ulrich bundles on geometrically ruled surfaces

  • Marian Aprodu
  • Gianfranco Casnati
  • Laura CostaEmail author
  • Rosa Maria Miró-Roig
  • Montserrat Teixidor I Bigas
Article
  • 7 Downloads

Abstract

We consider the following question: For which invariants g and e is there a geometrically ruled surface \(S \rightarrow C\) over a curve C of genus g with invariant e such that S is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper theta divisors on some moduli spaces of vector bundles on C with the existence of Ulrich line bundles on S will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.

Keywords

Vector bundle Ulrich bundle Geometrically ruled surface 

Mathematics Subject Classification

Primary 14J60 Secondary 14J26 

Notes

References

  1. 1.
    Aprodu, M., Costa, L., Miró-Roig, R.M.: Ulrich bundles on ruled surfaces. J. Pure Appl. Algebra 222, 131–138 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. Lond. Math. Soc. 7, 414–452 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beauville, A.: Determinantal hypersurfaces. Michigan Math. J. 48, 39–64 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Casanellas, M., Hartshorne, R., Geiss, F., Schreyer, F.O.: Stable Ulrich bundles. Int. J. Math. 23, 1250083 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Casnati, G.: Special Ulrich bundles on non-special surfaces with \(p_g=q=0\). Int. J. Math. 28, 1750061 (2017)CrossRefzbMATHGoogle Scholar
  6. 6.
    Casnati, G.: Ulrich bundles on non-special surfaces with \(p_g=0\) and \(q=1\). Rev. Mat. Complut. (2017).  https://doi.org/10.1007/s13163-017-0248-z zbMATHGoogle Scholar
  7. 7.
    Eisenbud, D., Harris, J.: 3264 & All That: A Second Course in Algebraic Geometry. Cambridge University Press, Cambridge (2016)CrossRefzbMATHGoogle Scholar
  8. 8.
    Eisenbud, D., Schreyer, F.-O., appendix by Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16, 537–579 (2003)Google Scholar
  9. 9.
    Faenzi, D., Malaspina, F.: Surfaces of minimal degree of tame representation type and mutations of Cohen–Macaulay modules. Adv. Math. 310, 663–695 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hartshorne, R.: Algebraic Geometry. G.T.M. 52, Springer (1977)Google Scholar
  11. 11.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd edn. Cambridge University Press, Cambridge Mathematical Library, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Nagata, M.: On self-intersection numbers of a section on a ruled surface. Nagoya Math. J. 37, 191–196 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Popa, M.: On the base locus of the generalized theta divisor. C. R. Acad. Sci. Paris Sér. I Math. 329(6), 507–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Raynaud, M.: Sections des fibrés vectoriels sur une courbe. Bull. Soc. Math. France 110, 103–125 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Russo, B., Teixidor, M.: On a conjecture of Lange. J. Algebraic Geom. 8, 483–496 (1999)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Teixidor, M.: Stable extensions by line bundles. J. Reine Angew. Math. 502, 163–173 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultatea de Matematică şi InformaticăUniversitatea din BucureştiBucureştiRomania
  2. 2.Institutul de Matematică “Simion Stoilow” al Academiei RomâneBucureştiRomania
  3. 3.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  4. 4.Facultat de Matemàtiques i Informàtica, Departament de Matemàtiques i InformàticaBarcelonaSpain
  5. 5.Mathematics DepartmentTufts UniversityMedfordUSA

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