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Rees algebras of additive group actions

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Abstract

We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme \(f:X\rightarrow S\) endowed with an action of the additive group scheme \({\mathbb {G}}_{a,S}\) over a base scheme or algebraic space S, which we call the (relative) Rees algebra of the \({\mathbb {G}}_{a,S}\)-action. In the case of affine algebraic varieties defined over a field of characteristic zero, we establish further properties of the Rees algebra of a \({\mathbb {G}}_{a}\)-action in terms of its associated locally nilpotent derivation. We give an algebro-geometric characterization of pairs consisting of an affine algebraic variety and a \({\mathbb {G}}_{a}\)-action on it whose associated Rees algebras are finitely generated and provide an algorithm extending van den Essen’s kernel algorithm for locally nilpotent derivations to compute generators of these Rees algebras. We illustrate these properties on several examples which played important historical roles in the development of the algebraic theory of locally nilpotent derivations and give applications to the construction of new families of affine threefolds with \({\mathbb {G}}_{a}\)-actions.

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References

  1. Alhajjar, B.: LND-Filtrations and Semi-Rigid Domains. arXiv:1501.00445

  2. Alhajjar, B.: Methods to compute ring invariants and applications: a new class of exotic threefolds. arXiv:1506.08522

  3. Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox Rings, Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge (2015)

    MATH  Google Scholar 

  4. Bourbaki, N.: Commutative algebra, Chapters 1–7. Translated from the French. Reprint of the: edition, p. 1989. Springer-Verlag, Berlin, Elements of Mathematics (Berlin) (1972)

  5. Deveney, K., Finston, R.: \({\mathbb{G}}_a\)-actions on \({\mathbb{C}}^3\) and \({\mathbb{C}}^7\). Commun. Algebra 22(15), 6295–6302 (1994)

    Article  Google Scholar 

  6. Deveney, J.K., Finston, D.R., Gehrke, M.: \({\mathbb{G}}_a\)-actions on \({\mathbb{C}}^n\). Commun. Algebra 22, 4977–4988 (1994)

    Article  Google Scholar 

  7. Dubouloz, A.: Danielewski–Fieseler surfaces. Transf. Groups 10(2), 139–162 (2005)

    Article  MathSciNet  Google Scholar 

  8. Dubouloz, A.: Exotic \({\mathbb{G}}_a\)-quotients of \(SL_2\times {\mathbb{A}}^1\). Eur. J. Math. 5(3), 828–844 (2019)

  9. Dubouloz, A., Fasel, J.: Families of \({\mathbb{A}}^1\)-contractible affine threefolds. Algebr. Geom. 5(1), 1–14 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Dubouloz, A., Finston, D.R.: On exotic affine 3-spheres. J. Algebraic Geom. 23(3), 445–469 (2014)

    Article  MathSciNet  Google Scholar 

  11. Dubouloz, A., Finston, D.R., Jaradat, I.: Proper triangular \({\mathbb{G}}_a\)-actions on \({\mathbb{A}}^4\) are translations. Algebra Number Theory 8(8), 1959–1984 (2014)

    Article  MathSciNet  Google Scholar 

  12. Dubouloz, A., Hedén, I., Kishimoto, T.: Equivariant extensions of \({mathbb{G}}_a\)-torsors over punctured surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XXI, 133–167 (2020)

    MATH  Google Scholar 

  13. Dubouloz, A., Poloni, P.-M.: On a class of Danielewski surfaces in affine 3-space. J. Algebra 321, 1797–1812 (2009)

    Article  MathSciNet  Google Scholar 

  14. van den Essen, A.: Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, vol. 190. Birkhäuser Verlag, Basel (2000)

    MATH  Google Scholar 

  15. Freudenburg, G.: Canonical factorization of the quotient morphism for an affine \({\mathbb{G}}_a\)-variety. Transform. Groups 24(2), 355–377 (2019)

    Article  MathSciNet  Google Scholar 

  16. Freudenburg, G.: Algebraic theory of locally nilpotent derivations, Second edition, Encyclopaedia of Mathematical Sciences, 136. VII. Springer, Berlin, Invariant Theory and Algebraic Transformation Groups (2017)

  17. Grothendieck, A.: Éléments de géométrie algébrique : II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. I.H.É.S. , Tome 8:5–222 (1961)

  18. Grothendieck, A.: Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, (Qautrième Partie). Publ. Math. I.H.É.S. , Tome 32, 5–360 (1968)

  19. Grosshans, F.D.: Contractions of the actions of reductive algebraic groups in arbitrary characteristic. Invent. Math. 107(1), 127–133 (1992)

    Article  MathSciNet  Google Scholar 

  20. Hartshorne, R.: Algebraic Geometry Graduate Texts in Mathematics Series, vol. 52. Springer, New York (1977)

    Google Scholar 

  21. Hedén, I.: Affine extensions of principal additive bundles over a punctured surface. Transform. Groups 21(2), 427–449 (2016)

    Article  MathSciNet  Google Scholar 

  22. Horrocks, G.: Vector bundles on the punctured spectrum of a local ring. Proc. London Math. Soc. (3) 14, 689–713 (1964)

    Article  MathSciNet  Google Scholar 

  23. Kaliman, S.: Free \({\mathbb{C}}_+\)-actions on \({\mathbb{C}}^3\) are translations. Invent. Math. 156(1), 163–173 (2004)

    Article  MathSciNet  Google Scholar 

  24. Kaliman, S.: Proper \({\mathbb{G}}_a\)-actions on \({\mathbb{C}}^4\) preserving a coordinate. Algebra Number Theory 12(2), 227–258 (2018)

    Article  MathSciNet  Google Scholar 

  25. Kaliman, S., Makar-Limanov, L.: On the Russell-Koras contractible threefolds. J. Algebra Geom. 6(2), 247–268 (1997)

    MathSciNet  MATH  Google Scholar 

  26. Kaliman, S., Makar-Limanov, L.: AK-Invariant of Affine Domains, Affine Algebraic Geometry, 231–255, Osaka University Press, Osaka (2007)

  27. Kaliman, S., Zaidenberg, M.: Affine modifications and affine hypersurfaces with a very transitive automorphism group. Transform. Groups 4(1), 53–95 (1999)

    Article  MathSciNet  Google Scholar 

  28. Lazarsfeld, R.: Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, p. Folge. 48. Springer, Berlin (2004)

  29. Laumon, G., Moret-Bailly, L.: Champs Algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 39. Springer, New York (2000)

    MATH  Google Scholar 

  30. Matsumura, H.: Commutative ring theory, Second edition. Cambridge Studies in Advanced Mathematics, p. 8. Cambridge University Press, Cambridge (1989)

  31. Miyanishi, M.: Lectures on curves on rational and unirational surfaces, Tata Institute of Fundamental Research. In: Lectures on mathematics and physics. Springer, Berlin (1978)

  32. Popov, V. L.: Contractions of actions of reductive algebraic groups (Russian); translated from Mat. Sb. (N.S.) 130(172) (1986), no. 3, 310–334, 431 Math. USSR-Sb. 58(2):311–335 (1987)

  33. Popp, H.: Moduli theory and classification theory of algebraic varieties. Lecture Notes in Mathematics, vol. 620. Springer-Verlag, Berlin-New York (1977)

  34. Rentschler, R.: Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267, 384–387 (1968)

    MathSciNet  MATH  Google Scholar 

  35. Seshadri, C.S.: Triviality of vector bundles over the affine space \(K^{2}\). Proc. Natl. Acad. Sci. USA 44, 456–458 (1958)

    Article  Google Scholar 

  36. Srinivas, V.: Some applications of algebraic cycles to affine algebraic geometry, Algebraic cycles, sheaves, shtukas, and moduli, 185–215. Trends Math, Birkhäuser, Basel (2008)

    Google Scholar 

  37. The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu (2017)

  38. Tanimoto, R.: An algorithm for computing the kernel of a locally finite iterative higher derivation. J. Pure Appl. Algebra 212(10), 2284–2297 (2008). https://doi.org/10.1016/j.jpaa.2008.03.006

    Article  MathSciNet  MATH  Google Scholar 

  39. Winkelmann, J.: On free holomorphic \({\mathbb{C}}_+\)-actions on \({\mathbb{C}}^n\) and homogeneous Stein manifolds. Math. Ann. 286, 593–612 (1990)

    Article  MathSciNet  Google Scholar 

  40. Zariski, O.: The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface. Ann. Math. (2) 76, 560–615 (1962)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the University of Saitama, at which this research was initiated during visits of the first and second authors, and the Institute of Mathematics of Burgundy, at which it was continued during a visit of the third author, for their generous support and the excellent working conditions offered. We also thank the referee for his thorough and detailed comments which contributed to improving readability.

The first author was partially supported by the French “Investissements d’Avenir” program, project ISITE-BFC MIAV ANR-15-IDEX-0008 and by ANR Project FIBALGA ANR-18-CE40-0003-01. The Institute of Mathematics of Burgundy receives support from the EIPHI Graduate School ANR-17-EURE-0002. The second author gratefully acknowledges support from the Knut and Alice Wallenberg Foundation, grant number KAW2016.0438 and Grant-in-Aid for JSPS Fellows No. 15F15751. The third author was partially funded by Grant-in-Aid for Scientific Research of JSPS No. 15K04805 and No. 19K03395.

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Authors and Affiliations

  1. IMB UMR5584, CNRS, Univ. Bourgogne Franche-Comté, Dijon, 21000, France

    Adrien Dubouloz

  2. Department of Mathematics, Uppsala University, Box 480, Uppsala, 751 06, Sweden

    Isac Hedén

  3. Department of Mathematics, Faculty of Science, Saitama University, Saitama, 338-8570, Japan

    Takashi Kishimoto

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  1. Adrien Dubouloz
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  2. Isac Hedén
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  3. Takashi Kishimoto
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Correspondence to Adrien Dubouloz, Isac Hedén or Takashi Kishimoto.

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Dubouloz, A., Hedén, I. & Kishimoto, T. Rees algebras of additive group actions. Math. Z. 301, 593–626 (2022). https://doi.org/10.1007/s00209-021-02926-0

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