Revista Matemática Complutense

, Volume 32, Issue 2, pp 559–574 | Cite as

Ulrich bundles on non-special surfaces with \(p_g=0\) and \(q=1\)

  • Gianfranco CasnatiEmail author


Let S be a surface with \(p_g(S)=0\), \(q(S)=1\) and endowed with a very ample line bundle \({\mathcal {O}}_S(h)\) such that \(h^1\big (S,{\mathcal {O}}_S(h)\big )=0\). We show that such an S supports families of dimension p of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large p. Moreover, we show that S supports stable Ulrich bundles of rank 2 if the genus of the general element in \(\vert h\vert \) is at least 2.


Vector bundle Ulrich bundle 

Mathematics Subject Classification

Primary 14J60 Secondary 14J26 14J27 14J28 


  1. 1.
    Andreatta, M., Sommese, A.: Classification of irreducible projective surfaces of smooth sectional genus \(\le 3\). Math. Scand. 67, 197–214 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aprodu, M., Costa, L., Miró-Roig, R.M.: Ulrich bundles on ruled surfaces. J. Pure Appl. Algebra 222, 131–138 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aprodu, M., Farkas, G., Ortega, A.: Minimal resolutions, Chow forms and Ulrich bundles on \(K3\) surfaces. J. Reine Angew. Math. 730, 225–249 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. London Math. Soc. 7, 414–452 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ballico, E., Chiantini, L.: Some properties of stable rank \(2\) bundles on algebraic surfaces. Forum Math. 4, 417–424 (1992)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beauville, A.: Complex Algebraic Surfaces. London Mathematical Society Student Texts 34. Cambridge University Press (1996)Google Scholar
  7. 7.
    Beauville, A.: Determinantal hypersurfaces. Mich. Math. J. 48, 39–64 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beauville, A.: Ulrich bundles on abelian surfaces. Proc. Am. Math. Soc. 144, 4609–4611 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beauville, A.: Ulrich bundles on surfaces with \(p_g=q=0\). arXiv:1607.00895 [math.AG]
  10. 10.
    Beauville, A.: An introduction to Ulrich bundles. Eur. J. Math. (2017).
  11. 11.
    Borisov, L., Nuer, H.: Ulrich bundles on Enriques surfaces. arXiv:1606.01459 [math.AG]
  12. 12.
    Casanellas, M., Hartshorne, R.: ACM bundles on cubic surfaces. J. Eur. Math. Soc. 13, 709–731 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Casanellas, M., Hartshorne, R., Geiss, F., Schreyer, F.O.: Stable Ulrich bundles. Int. J. Math. 23, 1250083 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Casnati, G.: Special Ulrich bundles on non-special surfaces with \(p_g=q=0\). Int. J. Math. 28, 1750061 (2017)CrossRefzbMATHGoogle Scholar
  15. 15.
    Casnati, G., Galluzzi, F.: Stability of rank \(2\) Ulrich bundles on projective \(K3\) surface. Math. Scand. (to appear). arXiv:1607.05469 [math.AG]
  16. 16.
    Casnati, G.: On the existence of Ulrich bundles on geometrically ruled surface. PreprintGoogle Scholar
  17. 17.
    Coskun, E., Kulkarni, R.S., Mustopa, Y.: The geometry of Ulrich bundles on del Pezzo surfaces. J. Algebra 375, 280–301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Coskun, E., Kulkarni, R.S., Mustopa, Y.: Pfaffian quartic surfaces and representations of Clifford algebras. Doc. Math. 17, 1003–1028 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Eisenbud, D., Schreyer, F.O., Weyman, J.: Resultants and Chow forms via exterior syzigies. J. Am. Math. Soc. 16, 537–579 (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Faenzi, D., Pons-Llopis, J.: The CM representation type of projective varieties. arXiv:1504.03819 [math.AG]
  21. 21.
    Gallego, F.J., Purnaprajna, B.P.: Normal presentation on elliptic ruled surfaces. J. Algebra 186, 597–625 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York, Heidelberg (1977)Google Scholar
  23. 23.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd edn. Cambridge University Press, Cambridge Mathematical Library, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Idà, M., Mezzetti, E.: Smooth non-special surfaces in \({\mathbb{P}}^4\). Manuscr. Math. 68, 57–77 (1990)CrossRefzbMATHGoogle Scholar
  25. 25.
    Mezzetti, E., Ranestad, K.: The non-existence of a smooth sectionally non-special surface of degree \(11\) and sectional genus \(8\) in the projective fourspace. Manuscr. Math. 70, 279–283 (1991)CrossRefzbMATHGoogle Scholar
  26. 26.
    Miró-Roig, R.M.: The representation type of rational normal scrolls. Rend. Circ. Mat. Palermo 62, 153–164 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Miró-Roig, R.M., Pons-Llopis, J.: Representation type of rational ACM surfaces \(X\subseteq {\mathbb{P}}^4\). Algebra Represent. Theor. 16, 1135–1157 (2013)CrossRefzbMATHGoogle Scholar
  28. 28.
    Miró-Roig, R.M., Pons-Llopis, J.: \(N\)-dimensional Fano varieties of wild representation type. J. Pure Appl. Algebra 218, 1867–1884 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mumford, D.: Lectures on Curves on an Algebraic Surface. With a section by G. M. Bergman. Annals of Mathematics Studies. Princeton University Press, Princeton (1966)Google Scholar
  30. 30.
    Nagata, M.: On self-intersection numbers of a section on a ruled surface. Nagoya Math. J. 37, 191–196 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pons-Llopis, J., Tonini, F.: ACM bundles on del Pezzo surfaces. Matematiche (Catania) 64, 177–211 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Serrano, F.: Divisors of bielliptic surfaces and embeddings in \({\mathbb{P}}^4\). Math. Z. 203, 527–533 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly

Personalised recommendations