Triangular 𝐺ₐ actions on 𝐂⁴

Deveney, James; Finston, David; van Rossum, Peter

doi:10.1090/S0002-9939-04-07500-8

Triangular $G_{a}$ actions on $\mathbf {C}^{4}$
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by James K. Deveney, David R. Finston and Peter van Rossum

Proc. Amer. Math. Soc. 132 (2004), 2841-2848

DOI: https://doi.org/10.1090/S0002-9939-04-07500-8

Published electronically: June 2, 2004

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Abstract:

Every locally trivial action of the additive group of complex numbers on four-dimensional complex affine space that is given by a triangular derivation is conjugate to a translation. A criterion for a proper action on complex affine $n$-space to be locally trivial is given, along with an example showing that the hypotheses of the criterion are sharp.

References

Arno van den Essen, An algorithm to compute the invariant ring of a $\textbf {G}_a$-action on an affine variety, J. Symbolic Comput. 16 (1993), no. 6, 551–555. MR 1279532, DOI 10.1006/jsco.1993.1062
Daniel Daigle and Gene Freudenburg, Triangular derivations of $\mathbf k[X_1,X_2,X_3,X_4]$, J. Algebra 241 (2001), no. 1, 328–339. MR 1838855, DOI 10.1006/jabr.2001.8763
V. I. Danilov, Rings with a discrete group of divisor classes, Mat. Sb. (N.S.) 88(130) (1972), 229–237 (Russian). MR 0306184
James K. Deveney, David R. Finston, and Mai Gehrke, $\textbf {G}_a$ actions on $\mathbf C^n$, Comm. Algebra 22 (1994), no. 12, 4977–4988. MR 1285720, DOI 10.1080/00927879408825115
James K. Deveney and David R. Finston, A proper $\textbf {G}_a$ action on $\textbf {C}^5$ which is not locally trivial, Proc. Amer. Math. Soc. 123 (1995), no. 3, 651–655. MR 1273487, DOI 10.1090/S0002-9939-1995-1273487-0
James K. Deveney and David R. Finston, $\mathbf G_a$ invariants and slices, Comm. Algebra 30 (2002), no. 3, 1437–1447. MR 1892608, DOI 10.1081/AGB-120004880
James K. Deveney and David R. Finston, Regular $G_a$ invariants, Osaka J. Math. 39 (2002), no. 2, 275–282. MR 1914293
Gert-Martin Greuel and Gerhard Pfister, Geometric quotients of unipotent group actions. II, Singularities (Oberwolfach, 1996) Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 27–36. MR 1652466
James K. Deveney and David R. Finston, Local triviality of proper $\textbf {G}_a$ actions, J. Algebra 221 (1999), no. 2, 692–704. MR 1728405, DOI 10.1006/jabr.1997.8028
David Wright, On the Jacobian conjecture, Illinois J. Math. 25 (1981), no. 3, 423–440. MR 620428
Herbert Popp, Moduli theory and classification theory of algebraic varieties, Lecture Notes in Mathematics, Vol. 620, Springer-Verlag, Berlin-New York, 1977. MR 0466143, DOI 10.1007/BFb0067436
Harald Holmann, Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann. 150 (1963), 327–360 (German). MR 150789, DOI 10.1007/BF01470762
Joseph Lipman, Unique factorization in complete local rings, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 531–546. MR 0374125
Jörg Winkelmann, On free holomorphic $\bf C$-actions on $\textbf {C}^n$ and homogeneous Stein manifolds, Math. Ann. 286 (1990), no. 1-3, 593–612. MR 1032948, DOI 10.1007/BF01453590
Elena Halanay, An example of a unique factorization domain whose completion is not a unique factorization domain, Stud. Cerc. Mat. 30 (1978), no. 5, 495–497 (Romanian, with English summary). MR 518249
P. van Rossum: Tackling Problems on Affine Space with Locally Nilpotent Derivations on Polynomial Rings, Thesis, Catholic University Nijmegen, 2001.
Robin Hartshorne and Arthur Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415–437. MR 347821, DOI 10.1080/00927877408548627
David Mumford, The red book of varieties and schemes, Second, expanded edition, Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999. Includes the Michigan lectures (1974) on curves and their Jacobians; With contributions by Enrico Arbarello. MR 1748380, DOI 10.1007/b62130
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531