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A Note on Locally Nilpotent Derivations and Variables of k[X, Y, Z]

Published online by Cambridge University Press:  20 November 2018

Daniel Daigle  and
Shulim Kaliman
Daniel Daigle
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5 e-mail: [email protected]
Shulim Kaliman
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL 33124, U.S.A. e-mail: [email protected]
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Abstract

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We strengthen certain results concerning actions of $\left( \mathbb{C},\,+ \right)$ on ${{\mathbb{C}}^{3}}$ and embeddings of ${{\mathbb{C}}^{2}}$ in ${{\mathbb{C}}^{3}}$, and show that these results are in fact valid over any field of characteristic zero.

Keywords

14R1014R2014R2513N15locally nilpotent derivationsgroup actionspolynomial automorphismsvariableaffine space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 4 , 01 December 2009 , pp. 535 - 543
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Bonnet, P., Surjectivity of quotient maps for algebraic (ℂ, +)-actions and polynomial maps with contractible fibers. Transform. Groups 7(2002), 3–14.Google Scholar
[2] Connell, E. H., Bass, H. and Wright, D. L., Locally polynomial algebras are symmetric algebras. Invent. Math. 38(1977), no. 3, 279–299.Google Scholar
[3] Eklof, P. C., Lefschetz's principle and local functors. Proc. Amer. Math. Soc. 37(1973), 333–339.Google Scholar
[4] Kaliman, S., Polynomials with general ℂ2 -fibers are variables. Pacific J. Math. 203(2002), no. 1, 161–190.Google Scholar
[5] Kaliman, S., Fre ℂ+-actions on ℂ3 are translations. Invent. Math. 156(2004), no. 1, 163–173.Google Scholar
[6] Kambayashi, T., On the absence of nontrivial separable forms of the affine plane. J. Algebra 35(1975), 449–456.Google Scholar
[7] Matsumura, H., Commutative Algebra. 2nd edition. Mathematics Lecture Note Series, Benjamin/Cummings, Reading, MA, 1980.Google Scholar
[8] Miyanishi, M., Normal affine subalgebras of a polynomial ring. In: Algebraic and Topological Theories. Kinokuniya, Tokyo, 1986, pp. 37–51.Google Scholar
[9] Ramanujam, C. P., A topological characterization of the affine plane as an algebraic variety. Ann. of Math. 94(1971), 69–88.Google Scholar
[10] Rentschler, R., Opérations du groupe additif sur le plane affine. C. R. Acad. Sci. Paris Sér A-B, 267(1968), A384–A387.Google Scholar
[11] Robinson, A., On theMetamathematics of Algebra. Studies in Logic and the Foundations of Mathematics, North-Holland Publishing, Amsterdam, 1951.Google Scholar
[12] Russell, P., Some Formal Aspects of the Theorems of Mumford-Ramanujam. Algebra, Arithmetic and Geometry, Mumbai 2000 Tata Institute of Fundamental Research, Narosa Publishing, 2002, pp. 557–584.Google Scholar
[13] Sathaye, A., Polynomial ring in two variables over a DVR: a criterion. Invent. Math. 74(1983), no. 1, 159–168.Google Scholar
[14] Varčenko, A. N., Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 36(1972), 957–1019, (English translation)Math. USSR-Izv., 6(1972), 949–1008.Google Scholar
[15] Weil, A., Foundation of Algebraic Geometry. Rev. ed. American Mathematical Society, Providence, RI, 1962.Google Scholar
[16] Wright, D., On the jacobian conjecture. Illinois J. Math. 25(1981), no. 3, 423–440.Google Scholar