Skip to main content
SpringerLink
Log in

A primer on toric varieties

  • Review Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

Abstract

These notes are based on mini-courses delivered at three summer schools: the Algebraic Geometry in Auckland in December 2019, the COW/EmSG/GLEN in September 2020 and the Lectures Sophie Kowalevski in June 2021. They are aimed at either graduate students or busy mathematicians in need either of a quick reminder, or of examples illustrating specific properties. There are no new results in these notes.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. If the reader enjoys the continued fraction algorithm in 2.4.2, there exists a construction of a distinguished resolution of a Gorenstein isolated threefold toric singularity [8] which uses continued fractions (this is related to the McKay correspondence).

  2. They also compute many numerical invariants of a toric surface with a given singularity content, and prove this content is itself invariant under mutation.

  3. Do not confuse this with the index d of \(\sigma \) in N.

  4. Here \(\sigma \) need not be a cone of the secondary fan itself, but only a cone generated by certain rays of this fan.

References

  1. Akhtar, M., Coates, T., Corti, A., Heuberger, L., Kasprzyk, A., Oneto, A., Petracci, A., Prince, T., Tveiten, K.: Mirror symmetry and the classification of orbifold del Pezzo surfaces. Proc. Amer. Math. Soc. 144(2), 513–527 (2016)

    Article  MathSciNet  Google Scholar 

  2. Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations. SIGMA 8, 94 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Akhtar, M., Kasprzyk, A.: Singularity content (2014). arXiv:1401.5458

  4. Beltrametti, M., Robbiano, L.: Introduction to the theory of weighted projective spaces. Expo. Math. 4(2), 111–162 (1986)

    MathSciNet  MATH  Google Scholar 

  5. Brasselet, J.-P.: Introduction to Toric Varieties. In: 23rd Brazilian Mathematics Colloquium, Rio de Janeiro, Brazil, July 22–27, 2001. IMPA Mathematical Publications. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2001)

  6. Corti, A., Filip, M., Petracci, A.: Mirror symmetry and smoothing Gorenstein toric affine 3-folds. In: Aluffi, P. et al. (eds.) Facets of Algebraic Geometry, vol. I. London Mathematical Society Lecture Note Series, vol. 472, pp. 132–163. Cambridge University Press, Cambridge (2022)

  7. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies of Mathematics, vol. 124. American Mathematical Society, Providence (2011)

  8. Craw, A., Reid, M.: How to calculate \(A\)-Hilb \({\mathbb{C}}^3\). In: Bonavero, L., Brion, M. (eds.) Geometry of Toric Varieties. Séminaires et Congrès, vol. 6, pp. 129–154. Société Mathématique de France, Paris (2002)

    Google Scholar 

  9. Dolgachev, I.: Weighted projective varieties. In: Carrell, J.B. (ed.) Group Actions and Vector Fields, Lecture Notes in Mathematics, vol. 956, pp. 34–71. Springer, Berlin (1982)

    Chapter  Google Scholar 

  10. Fujino, O., Sato, H.: Introduction to the toric Mori theory. Mich. Math. J. 52(3), 649–665 (2004)

    Article  MathSciNet  Google Scholar 

  11. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. The 1989 William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)

  12. García, J.M.: Toric Varieties in a Nutshell. http://jesusmartinezgarcia.net/wp-content/uploads/2019/08/toricNutshell.pdf

  13. Iano-Fletcher, A.R.: Working with weighted complete intersections. In: Corti, A., Reid, M. (eds.) Explicit Birational Geometry of 3-Folds, pp. 101–173. Cambridge University Press, Cambridge (2000)

    Chapter  Google Scholar 

  14. Kalashnikov, E.: Mirror symmetry for GIT quotients and their subvarieties. https://scholar.harvard.edu/files/elanakalashnikov/files/short_version.pdf

  15. Kollár, J., Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)

    Article  MathSciNet  Google Scholar 

  16. Petracci, A.: On deformations of toric Fano varieties (2019) (to appear in Interactions with Lattice Polytopes). arXiv:1912.01538

  17. Telen, S.: Introduction to toric geometry (2022). arXiv:2203.01690

  18. Wiśniewski, J.A.: Toric Mori theory and Fano manifolds. In: Bonavero, L., Brion, M. (eds.) Geometry of Toric Varieties. Séminaires et Congrès, vol. 6, pp. 249–272. Société Mathématique de France, Paris (2002)

    MATH  Google Scholar 

Download references

Acknowledgements

I warmly thank the organisers of the three summer schools: the Algebraic Geometry in Auckland in December 2019, the COW/EmSG/GLEN in September 2020 and the Lectures Sophie Kowalevski in June 2021, and in particular Ivan Cheltsov who first invited me to give this talk series and encouraged me along the way. I am very grateful to Jean-Baptiste Campesato and Marco Golla for having proofread drafts of these notes. The referee of this note has been particularly helpful and thorough, which I greatly appreciated. I thank Alessio Corti and Andrea Petracci, who throughout many years have enlightened me on various aspects of toric geometry. Last but not least, I thank the students that attended these talks, whose feedback I have found very motivating.

Author information

Authors and Affiliations

  1. Laboratoire Angevin de Recherche en Mathématiques, Université d’Angers, 2 Boulevard Lavoisier, 49045, Angers Cedex 01, France

    Liana Heuberger

Authors
  1. Liana Heuberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liana Heuberger.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heuberger, L. A primer on toric varieties. European Journal of Mathematics 8, 952–971 (2022). https://doi.org/10.1007/s40879-022-00561-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-022-00561-5

Keywords

Mathematics Subject Classification

Search

Navigation