Skip to main content
SpringerLink
Log in

Segre classes on smooth projective toric varieties

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We provide a generalization of the algorithm of Eklund, Jost and Peterson for computing Segre classes of closed subschemes of projective \(k\)-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.

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

Fig. 1

Similar content being viewed by others

References

  1. Aluffi, P.: Computing characteristic classes of projective schemes. J. Symbol. Comput. 35(1), 3–19 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Algebr. Geom. 4(1), 17–50 (1995)

    MATH  Google Scholar 

  3. Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties. In: Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)

  4. Danilov, V.I.: The geometry of toric varieties. Uspekhi Mat. Nauk 33, 2(200) 85–134, 247 (1978)

  5. Eklund, D., Jost, C., Peterson, C.: A method to compute Segre classes of subschemes of projective space. J. Algebra Appl. (2011, to appear). arXiv:1109.5895 [math.AG]

  6. Fulton, W.: Intersection theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2. Springer, Berlin (1984)

  7. Fulton, W.: Introduction to toric varieties. In: Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

  8. Grayson, D.R., Stillman, M.E.: Macaulay2. A software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/

  9. Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, No. 52. Springer, New York (1977)

  10. Maclagan, D., Smith, G.G.: Multigraded Castelnuovo-Mumford regularity. J. Reine Angew. Math. 571, 179–212 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Peskine, C., Szpiro, L.: Liaison des variétés algébriques. I. Invent. Math. 26, 271–302 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  12. Stein, W. A., et al.: Sage Mathematics Software (Version 4.8). The Sage Development Team (2012). http://www.sagemath.org

Download references

Acknowledgments

We want to thank Ragni Piene for valuable supervision, Christine Jost for comments and for pointing us towards the Sage [12] implementation of intersection theory in the toric setting, and Kristian Ranestad for reminding us of the importance of nefness. Furthermore, we are grateful to Carel Faber and the referee for constructive remarks and comments. Finally, we want to thank Terje Kvernes at Drift and Georg Muntingh for computer assistance. All computations are performed using Macaulay2 [8] by Grayson and Stillman with the module NormalToricVarieties by Smith, and Sage [12] by Stein et al.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 , Oslo, Norway

    Torgunn Karoline Moe & Nikolay Qviller

Authors
  1. Torgunn Karoline Moe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolay Qviller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Torgunn Karoline Moe.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moe, T.K., Qviller, N. Segre classes on smooth projective toric varieties. Math. Z. 275, 529–548 (2013). https://doi.org/10.1007/s00209-013-1146-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-013-1146-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation