Proper twin-triangular 𝔾ₐ-actions on 𝔸⁴ are translations

Dubouloz, Adrien; Finston, David

doi:10.1090/S0002-9939-2014-11932-0

Proper twin-triangular $\mathbb {G}_{a}$-actions on $\mathbb {A}^{4}$ are translations
HTML articles powered by AMS MathViewer

by Adrien Dubouloz and David R. Finston PDF

Proc. Amer. Math. Soc. 142 (2014), 1513-1526 Request permission

Abstract:

An additive group action on an affine $3$-space over a complex Dedekind domain $A$ is said to be twin-triangular if it is generated by a locally nilpotent derivation of $A[y,z_{1},z_{2}]$ of the form $r\partial _{y}+p_{1}(y)\partial _{z_{1}}+p_{2}(y)\partial _{z_{2}}$, where $r\in A$ and $p_{1},p_{2}\in A[y]$. We show that these actions are translations if and only if they are proper. Our approach avoids the computation of rings of invariants and focuses more on the nature of geometric quotients for such actions.

References

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Hyman Bass, A nontriangular action of $\textbf {G}_{a}$ on $\textbf {A}^{3}$, J. Pure Appl. Algebra 33 (1984), no. 1, 1–5. MR 750225, DOI 10.1016/0022-4049(84)90019-7
H. Bass, E. H. Connell, and D. L. Wright, Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1976/77), no. 3, 279–299. MR 432626, DOI 10.1007/BF01403135
Daniel Daigle and Gene Freudenburg, Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of $k[X_1,\cdots ,X_n]$, J. Algebra 204 (1998), no. 2, 353–371. MR 1624439, DOI 10.1006/jabr.1998.7465
James K. Deveney and David R. Finston, $G_a$ actions on $\textbf {C}^3$ and $\textbf {C}^7$, Comm. Algebra 22 (1994), no. 15, 6295–6302. MR 1303005, DOI 10.1080/00927879408825190
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, Regular $G_a$ invariants, Osaka J. Math. 39 (2002), no. 2, 275–282. MR 1914293
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, David R. Finston, and Peter van Rossum, Triangular $\mathbf G_a$ actions on $\mathbf C^4$, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2841–2848. MR 2063101, DOI 10.1090/S0002-9939-04-07500-8
A. Dubouloz, Sur une classe de schémas avec actions de fibrés en droites (on a class of schemes with line bundles actions), Ph.D. thesis, Université Joseph-Fourier, 2004.
A. Dubouloz, Affine open subsets in $\mathbb {A}^3$ without the cancellation property, Commutative Algebra and Algebraic Geometry (CAAG-2010): Proceedings of the Bangalore conference in honour of B. Singh, U. Storch and R. Gurjar, Ramanujan Mathematical Society Lecture Notes Series, no. 17, 2013. (ISBN: 9789380416106)
Adrien Dubouloz and Pierre-Marie Poloni, On a class of Danielewski surfaces in affine 3-space, J. Algebra 321 (2009), no. 7, 1797–1812. MR 2494748, DOI 10.1016/j.jalgebra.2008.12.009
Karl-Heinz Fieseler, On complex affine surfaces with $\textbf {C}^+$-action, Comment. Math. Helv. 69 (1994), no. 1, 5–27. MR 1259603, DOI 10.1007/BF02564471
Shulim Kaliman, Free $\mathbf C_+$-actions on $\mathbf C^3$ are translations, Invent. Math. 156 (2004), no. 1, 163–173. MR 2047660, DOI 10.1007/s00222-003-0336-1
Herbert Popp, Moduli theory and classification theory of algebraic varieties, Lecture Notes in Mathematics, Vol. 620, Springer-Verlag, Berlin-New York, 1977. MR 0466143
Rudolf Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A384–A387 (French). MR 232770
A. Sathaye, Polynomial ring in two variables over a DVR: a criterion, Invent. Math. 74 (1983), no. 1, 159–168. MR 722731, DOI 10.1007/BF01388536
Dennis M. Snow, Triangular actions on $\textbf {C}^3$, Manuscripta Math. 60 (1988), no. 4, 407–415. MR 933471, DOI 10.1007/BF01258660
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