Cancellation

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 79)

Abstract

What follows is a slightly expanded and updated version of lectures I gave in May 2011 during a workshop on “Group actions, generalized cohomology theories and affine algebraic geometry” at the University of Ottawa. Among the participants were young beginning mathematicians as well as seasoned experts in diverse aspects of algebra and geometry. The aim (by order of the organizers) was to give all of them a taste of “cancellation for affine algebraic varieties.”As much as possible I have tried in these notes to maintain the informal style of the lectures. They are a very selective and far from an exhaustive treatment of the subject. Should the reader note a tendency to frequently switch between the algebraic and geometric point of view: this is by design.

Keywords

Projective Module Local Equation Effective Divisor Kodaira Dimension Irreducible Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This research was supported by a grant from NSERC, Canada.

References

  1. 1.
    T. Asanuma, Polynomial fibre rings of algebras over noetherian rings. Invent. Math. 87, 101–127 (1987)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    T. Asanuma, Non-linearizable algebraic group actions on \(\mathbb{A}^{n}\). J. Algebra 166(1), 72–79 (1994)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    S. Abhyankar, P. Eakin, W. Heinzer, On the uniqueness of the ring of coefficients in a polynomial ring. J. Algebra 23, 310–342 (1972)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    S.S. Abhyankar, T.-T. Moh, Embeddings of the line in the plane. J. Reine Angew. Math. 276, 148–166 (1975)MATHMathSciNetGoogle Scholar
  5. 5.
    A. Beauville, J.-L. Colliot-Thelene, J.-J. Sansuc, P. Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles. Ann. Math. (2) 121(2), 283–318 (1985)Google Scholar
  6. 6.
    R. Bott, J. Milnor, On the parallelizability of the spheres. Bull. Am. Math. Soc. 64, 87–89 (1958)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    J. Brun, On the cancellation problem for compact analytic manifolds. Proc. Symp. Pure Math. 30, 245–247 (1976)Google Scholar
  8. 8.
    J. Brun, Sur la simplification dans les ismorphismes analytiques. Ann. Ec. Norm. Sup. 4(serie 9), 533–538 (1976)Google Scholar
  9. 9.
    A. Crachiola, L. Makar-Limanov, An algebraic proof of a cancellation theorem for surfaces. J. Algebra 320, 3113–3119 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    A. Dimca, in Topology and Singularities of Hypersurfaces. Universitext (Springer, New York, 1992)Google Scholar
  11. 11.
    I. Dolgachev, B. Weisfeiler, Unipotent group schemes over integral rings (English translation). Izv. Akad. Nauk USSR Ser. Mat. 38, 757–799 (1974)MATHGoogle Scholar
  12. 12.
    A. Dubouloz, L. Moser-Jauslin, P.-M. Poloni, Noncancellation for contractible affine threefolds. Proc. Am. Math. Soc. 139(12), 4273–4284 (2011)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    P. Eakin, W. Heinzer, A cancellation problem for rings. in Conference on Commutative Algebra. Lecture Notes in Mathematics, vol. 311 (Springer, Berlin, 1973), pp. 61–77Google Scholar
  14. 14.
    K.H. Fieseler, On complex affine surfaces with \(\mathbb{C}_{+}\)-action. Math. Helvetici 69, 5–27 (1994)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    T. Fujita, On Zariski problem. Proc. Jpn. Acad. Ser. A 55(3), 106–110 (1979)CrossRefMATHGoogle Scholar
  16. 16.
    T. Fujita, Cancellation problem of complete varieties. Invent. Math. 64, 119–121 (1981)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    T. Fujita, On the topology of non-complete surfaces. J. Fac. Sci. U. Tokyo 29, 503–566 (1982)MATHGoogle Scholar
  18. 18.
    T. Fujita, S. Iitaka, Cancellation theorem for algebraic varieties. J. Fac. Sci. U. Tokyo 24, 123–127 (1977)MATHMathSciNetGoogle Scholar
  19. 19.
    N. Gupta, On the cancellation problem for the affine space \(\mathbb{A}^{3}\) in characteristc p. Inventiones Mat. 195, 279–288 (2014)Google Scholar
  20. 20.
    N. Gupta, On the family of affine theefolds x m y = F(x, z, t), to appear in Compositio Mat.Google Scholar
  21. 21.
    M. Hochster, Non-uniqueness of the ring of coefficients in a polynomial ring. Proc. Am. Math. Soc. 34, 81–82 (1972)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, in Complex Analysis and Algebraic Geometry (Iwanami Shoten, Tokyo, 1977), pp. 175–189Google Scholar
  23. 23.
    S. Kaliman, L. Makar Limanov, On the Russell-Koras contractible threefolds. J. Algebr. Geom. 6, 247–248 (1997)MATHMathSciNetGoogle Scholar
  24. 24.
    S. Kaliman, M. Koras, L. Makar-Limanov, P. Russell, \(\mathbb{C}^{{\ast}}\)-actions on \(\mathbb{C}^{3}\) are linear. Electron. Res. Announc. Am. Math. Soc. 3, 63–71 (1997)Google Scholar
  25. 25.
    S. Kaliman, S. Vénéreau, M. Zaidenberg, Simple birational extensions of the polynomial algebra \(\mathbb{C}^{[3]}\). Trans. Am. Math. Soc. 356(2), 509–555 (2003)CrossRefGoogle Scholar
  26. 26.
    M. Koras, P. Russell, Contractible threefolds and \(\mathbb{C}^{{\ast}}\)-actions on \(\mathbb{C}^{3}\). J. Algebr. Geom. 6, 671–695 (1997)MATHMathSciNetGoogle Scholar
  27. 27.
    M. Koras, P. Russell, \(\mathbb{C}^{{\ast}}\)-actions on \(\mathbb{C}^{3}\): the smooth locus of the quotient is not of hyperbolic type. J. Algebr. Geom. 8(4), 603–694 (1999)Google Scholar
  28. 28.
    L. Makar-Limanov, On groups of automorphisms of a class of surfaces. Isr. J. Math. 69(2), 250–256 (1990)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    L. Makar-Limanov, On the hypersurface \(x + x^{2}y + z^{2} + t^{3} = 0\) in \(\mathbb{C}^{4}\) or a \(\mathbb{C}^{3}\)-like threefold which is not \(\mathbb{C}^{3}\). Isr. J. Math. 96(Part B), 419–429 (1996)Google Scholar
  30. 30.
    M. Miyanishi, T. Sugie, Affine surfaces containing cylinderlike open sets. J. Math. Kyoto U. 20, 11–42 (1980)MATHMathSciNetGoogle Scholar
  31. 31.
    D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes tudes Sci. Publ. Math. 9, 5–22 (1961)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    C.P. Ramanujam, A topological characterization of the affine plane as an algebraic variety. Ann. Math. 94, 68–88 (1971)CrossRefMathSciNetGoogle Scholar
  33. 33.
    P. Russell, On affine-ruled surfaces. Math. Ann. 255, 287–302 (1981)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    B. Segre, Corrispondenze di Möbius e transformazioni cremoniane intere. Atti Acad. Sci. Torino Cl. Fis. Mat. Nat. 91, 319 (1956/1957)Google Scholar
  35. 35.
    T. Shioda, Some remarks on abelian varieties. J. Fac. Sci. U. Tokyo 24, 11–21 (1977)MATHMathSciNetGoogle Scholar
  36. 36.
    V. Shpilrain, J.-T. Yu, Affine varieties with equivalent cylinders. J. Algebra 251, 295–307 (2002)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace \(\mathbb{C}^{2}\). J. Math. Soc. Jpn. 26, 241–257 (1974)CrossRefMATHGoogle Scholar
  38. 38.
    J. Wilkens, On the cancellation problem for surfaces. C. R. Acad. Sci. Paris 326(Serie I), 1111–1116 (1998)Google Scholar
  39. 39.
    M. Zaidenberg, On exotic algebraic structures on affine spaces (Russian). Algebra i Analiz 19(5), 3–73 (1999) [English translation in St.Petersburg Math. J. 11(5), 703–760 (2000)]Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsMcGill UniversityMontrealCanada

Personalised recommendations