Skip to main content
SpringerLink
Log in

Instanton sheaves on Fano threefolds

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Generalizing the definitions originally presented by Kuznetsov and Faenzi, we study (possibly non locally free) instanton sheaves of arbitrary rank on Fano threefolds. We classify rank 1 instanton sheaves and describe all curves whose structure sheaves are rank 0 instanton sheaves. In addition, we show that every rank 2 instanton sheaf is an elementary transformation of a locally free instanton sheaf along a rank 0 instanton sheaf. To complete the paper, we describe the moduli space of rank 2 instanton sheaves of charge 2 on a quadric threefold X and show that the full moduli space of rank 2 semistable sheaves on X with Chern classes \((c_1,c_2,c_3)=(-\,1,2,0)\) is connected and contains, besides the instanton component, just one other irreducible component which is also fully described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

I hereby declare that data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

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

    Article  MathSciNet  Google Scholar 

  2. Atiyah, M., Drinfeld, V., Hitchin, N., Manin, Yu.: Construction of instantons. Phys. Lett. A65(3), 185–187 (1978)

    Article  MathSciNet  Google Scholar 

  3. Antonelli, V., Casnati, G.: Instanton sheaves on projective schemes. J. Pure Appl. Algebra 227, 107246 (2023). (40 pp)

    Article  MathSciNet  Google Scholar 

  4. Antonelli, V., Malaspina, F.: \(H\)-instanton bundles on three-dimensional polarized projective varieties. J. Algebra 598, 570–607 (2022)

    Article  MathSciNet  Google Scholar 

  5. Banica, C., Forster, O.: Multiplicity structures on space curves. Contemp. Math. 58, 47–64 (1986)

    MathSciNet  Google Scholar 

  6. Brambilla, M.C., Faenzi, D.: Vector bundles on Fano threefolds of genus 7 and Brill–Noether loci. Int. J. Math. 25, 1450023 (2014)

    Article  MathSciNet  Google Scholar 

  7. Casnati, G., Coskun, E., Genç, Ö., Malaspina, F.: Instanton bundles on the blow-up of the projective 3-space at a point. Mich. Math. J. 70, 807–836 (2021)

    Article  Google Scholar 

  8. Costa, L., Ottaviani, G.: Nondegenerate multidimensional matrices and instanton bundles. Trans. Am. Math. Soc. 355, 49–55 (2003)

    Article  MathSciNet  Google Scholar 

  9. 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. 19, 985–1004 (2000)

    Article  Google Scholar 

  10. Ein, L., Sols, I.: Stable vector bundles on quadric hypersurfaces. Nagoya Math. J. 96, 11–22 (1984)

    Article  MathSciNet  Google Scholar 

  11. Faenzi, D.: Even and odd instanton bundles on Fano threefolds of Picard number one. Manuscr. Math. 144, 199–239 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gargate, M., Jardim, M.: Singular loci of instanton sheaves on projective space. Int. J. Math. 27, 1640006 (2016)

    Article  MathSciNet  Google Scholar 

  13. Hartshorne, R.: Stable vector bundles of rank 2 on \({\mathbb{P} }^3\). Math. Ann. 238, 229–280 (1978)

    Article  MathSciNet  Google Scholar 

  14. Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254, 121–176 (1980)

    Article  MathSciNet  Google Scholar 

  15. Hauzer, M., Langer, A.: Moduli spaces of framed perverse instantons on \({\mathbb{P} }^3\). Glasg. Math. J. 53, 51–96 (2011)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  17. Iskovskih, V.A., Prokhorov, Yu.G.: Algebraic Geometry V: Fano varieties. Encyclopaedia of Mathematical Sciences, vol. 47. Springer-Verlag, Berlin (1999)

    Google Scholar 

  18. Jardim, M.: Instanton sheaves on complex projective spaces. Collect. Math. 57(1), 69–91 (2006)

    MathSciNet  Google Scholar 

  19. Jardim, M., Verbitsky, M.: Trihyperkähler reduction and instanton bundles on \({\mathbb{C} }{\mathbb{P} }^3\). Compos. Math. 150, 1836–1868 (2014)

    Article  MathSciNet  Google Scholar 

  20. Jardim, M., Maican, M., Tikhomirov, A.S.: Moduli spaces of rank 2 instanton sheaves on the projective space. Pac. J. Math. 291, 399–424 (2017)

    Article  MathSciNet  Google Scholar 

  21. Jardim, M., Markushevich, D., Tikhomirov, A.S.: Two infinite series of moduli spaces of rank 2 sheaves on \({\mathbb{P} }^3\). Ann. Mat. Pura Appl. 196, 1573–1608 (2017)

    Article  MathSciNet  Google Scholar 

  22. Jardim, M., Markushevich, D., Tikhomirov, A.S.: New divisors in the boundary of the instanton moduli space. Mosc. Math. J. 18, 117–148 (2018)

    MathSciNet  Google Scholar 

  23. Kuznetsov, A.: Instanton bundles on Fano threefolds. Cent. Eur. J. Math. 10, 1198–1231 (2012)

    Article  MathSciNet  Google Scholar 

  24. Lange, H.: Universal families of extensions. J. Algebra 83, 101–112 (1983)

    Article  MathSciNet  Google Scholar 

  25. Mamone Capria, M., Salamon, S.M.: Yang-Mills fields on quaternionic spaces. Nonlinearity 1, 517–530 (1988)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  27. Maruyama, M.: Boundedness of semistable sheaves of small ranks. Nagoya Math. J. 78, 65–94 (1980)

    Article  MathSciNet  Google Scholar 

  28. Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Natl. Acad. Sci. 86, 3000–3002 (1989)

    Article  MathSciNet  Google Scholar 

  29. Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Birkhauser, Basel (1980)

    Book  Google Scholar 

  30. Ottaviani, G.: Spinor bundles on quadrics. Trans. Am. Math. Soc. 307, 301–316 (1988)

    Article  MathSciNet  Google Scholar 

  31. Ottaviani, G., Szurek, M.: On moduli of stable 2-bundles with small Chern classes on \(Q3\). Ann. Mat. Pura Appl. 167, 191–241 (1994)

    Article  MathSciNet  Google Scholar 

  32. Tikhomirov, A.S.: Moduli of mathematical instanton vector bundles with odd \(c_2\) on projective space. Izv. Math. 76, 991–1073 (2012)

    Article  MathSciNet  Google Scholar 

  33. Tikhomirov, A.S.: Moduli of mathematical instanton vector bundles with even \(c_2\) on projective space. Izv. Math. 77, 1195–1223 (2013)

    Article  MathSciNet  Google Scholar 

  34. Qin, X.: Compactification of the moduli space of minimal instantons on the Fano 3-fold \(V_5\). J. Pure Appl. Algebra 225, 106526 (2021)

    Article  Google Scholar 

  35. Qin, X.: Compactification of the moduli space of minimal instantons on the Fano threefold \(V_4\). Eur. J. Math. 7, 1502–1523 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

MJ is supported by the CNPQ Grant Number 305601/2022-9 and the FAPESP Thematic Project 2018/21391-1. GC was supported by the FAPESP post-doctoral Grant Number 2019/21140-1.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Estatística e Computação Científica, Instituto de Matemática, Universidade Estadual de Campinas (UNICAMP), Rua Sérgio Buarque de Holanda, 651, Campinas, SP, 13083-970, Brazil

    Gaia Comaschi & Marcos Jardim

Authors
  1. Gaia Comaschi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcos Jardim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcos Jardim.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Comaschi, G., Jardim, M. Instanton sheaves on Fano threefolds. manuscripta math. (2024). https://doi.org/10.1007/s00229-024-01559-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00229-024-01559-x

Mathematics Subject Classification

Search

Navigation