Skip to main content
SpringerLink
Log in

EQUIVARIANT COX RING

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The ordinary Cox ring is also finitely generated and inherits a canonical structure of U-algebra. Using a work of Hausen and Herppich, we prove that the subalgebra of U-invariants of the Cox ring is a finitely generated Cox ring of a variety of complexity one under the action of a torus. As an application, we provide a criterion of combinatorial nature for the Cox ring of an almost homogeneous G-variety of complexity one to have log terminal singularities. Finally, we prove that for a normal rational G-variety of complexity one satisfying a mild additional condition (e.g., complete or almost homogeneous), the iteration of Cox rings is finite.

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

References

  1. Arzhantsev, I., Braun, L., Hausen, J., Wrobel, M.: Log terminal singularities, Platonic tuples and iteration of Cox rings. Europ. J. Math. 4(1), 242–312 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox Rings, Cambridge Studies in Advanced Mathematics, Vol. 144, Cambridge University Press, Cambridge, 2015.

  3. Bass, H., Haboush, W.: Linearizing certain reductive group actions. Trans. Amer. Math. Soc. 292(2), 463–482 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berchtold, F., Hausen, J.: Homogeneous coordinates for algebraic varieties. J. Algebra. 266(2), 636–670 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brion, M., Luna, D., Vust, T.: Espaces homogènes sphériques. Invent. Math. 84, 617–632 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Braun, J. Moraga, Iteration of Cox rings of klt singularities, arXiv:2103. 13524 (2021).

  7. L. Braun, Gorensteinness and iteration of Cox rings for Fano type varieties, arXiv:1903.07996 (2019).

  8. Brion, M.: The total coordinate ring of a wonderful variety. J. Algebra. 313(1), 61–99 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Brion, Linearization of algebraic group actions, in: Handbook of Group Actions, Vol. IV, Adv. Lect. Math. (ALM), Vol. 41, Int. Press, Somerville, MA, 2018, pp. 291–340.

  10. Edidin, D., Graham, W.: Equivariant intersection theory. Invent. Math. 131, 595–634 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné), Publ. Math. IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).

  12. B. Fantechi, L. Gttsche, L. Illusie, S. Kleiman, N. Nitsure, A. Vitsoli, Fundamental Algebraic Geometry, Grothendieck’s FGA Explained, Mathematical Surveys and Monographs, Vol. 123, AMS, Providence, RI, 2005.

  13. Flenner, H., Zaidenberg, M.: Normal affine surfaces with ℂ*-action. Osaka J. Math. 40(4), 981–1009 (2003)

    MathSciNet  MATH  Google Scholar 

  14. G. Gagliardi, Luna–Vust invariants of Cox rings of spherical varieties, arXiv: 1608.08151 (2016).

  15. Gongyo, Y., Okawa, S., Sannai, A., Takagi, S.: Characterization of varieties of Fano type via singularities of Cox rings. J. Algebr. Geom. 24(1), 159–182 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hartshorne, R.: Generalized divisors on Gorenstein schemes. K-Theory. 8(3), 287–339 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer, Berlin, 1977.

  18. M. Hashimoto, Equivariant class group III, Almost principal fiber bundles, arXiv:1503.02133 (2015).

  19. Hu, Y., Keel, S.: Mori Dream Spaces and GIT. Mich. Math. J. 48(1), 331–348 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hausen, J., Wrobel, M.: On iteration of Cox rings. J. Pure Appl. Algebra. 222(9), 2737–2745 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ishii, S.: Introduction to Singularities. Springer, Berlin (2018)

    Book  MATH  Google Scholar 

  22. J. Kollar, Lectures on Resolution of Singularities, Annals of Mathematics Studies, Vol. 166, Princeton University Press, Princeton, 2007.

  23. Knop, F.: Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheiteins. Math. Z. 213(1), 33–35 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Knop, F.: ber Bewertungen, welche unter einer reduktiven Gruppe invariant sind. Math. Ann. 295, 333–363 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Martin, Automorphism schemes of (quasi-)bielliptic surfaces, arXiv:2008. 11679 (2020).

  26. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  27. J. S. Milne, Algebraic Groups, the Theory of Group Schemes of Finite Type over a Field, Cambridge Studies in Advanced Mathematics, Vol. 170, Cambridge University Press, Cambridge, 2017.

  28. Е. В. Пономарёва, Инварианты коксовых колец двух хлопьев гov мало времени классических групп, Тр. ММО 86 (2015). hom. 1, 85–150. Engl. transl.: E. V. Ponomareva, Invariants of the Cox rings of low-complexity double flag varieties for classical groups, Trans. Moscow Math. Soc. 76 (2015), no. 1, 71–133.

  29. Petersen, L., Süss, H.: Torus invariant divisors. Israel J. Math. 182, 481–504 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ. 14(1), 1–28 (1974)

    MathSciNet  MATH  Google Scholar 

  31. Revêtements Étales et Groupe Fondamental, Séminaire de géométrie algébrique du Bois Marie 1960–61 dirigé par A. Grothendieck, Documents Mathématiques 3, Soc. Math. France, Paris, 2003.

  32. Schémas en Groupes, Séminaire de géométrie algébrique du Bois Marie 1962–64 dirigé par M. Demazure et A. Grothendieck, Documents Mathématiques 8, Soc. Math. France, Paris, 2011.

  33. D. A. Timashev, Homogeneous Spaces and Equivariant Embeddings, Encyclopaedia of Mathematical Sciences, Vol. 138, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. VIII, Springer, Berlin, 2011.

  34. A. Vezier, Cox rings of almost homogeneous SL2-threefolds, arXiv:2009.08676 (2020).

Download references

Author information

Authors and Affiliations

  1. Université Grenoble Alpes, Institut Fourier, CS 40700, 38058, Grenoble Cedex 9, France

    A. VEZIER

Authors
  1. A. VEZIER
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. VEZIER.

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

VEZIER, A. EQUIVARIANT COX RING. Transformation Groups 28, 1721–1774 (2023). https://doi.org/10.1007/s00031-022-09720-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-022-09720-0

Search

Navigation