On automorphisms of Danielewski surfaces

Crachiola, Anthony

doi:10.1090/S1056-3911-05-00414-5

Author: Anthony J. Crachiola
Journal: J. Algebraic Geom. 15 (2006), 111-132
DOI: https://doi.org/10.1090/S1056-3911-05-00414-5
Published electronically: May 12, 2005
MathSciNet review: 2177197
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Abstract | References | Additional Information

Abstract: Let $\mathbf {k}$ be any field. Let $R = \mathbf {k}[X,Y,Z]/(X^n Y - Z^2 - h(X)Z)$, where $h(0) \ne 0$ and $n \geq 2$. We develop techniques for computing the AK invariant of a domain with arbitrary characteristic. We use these techniques to compute $\operatorname {AK}(R)$, describe the automorphism group of $R$, and describe the isomorphism classes of these algebras. We then show that these algebras provide counterexamples to the cancellation problem over any field, extending Danielewski’s original counterexample.

Shreeram S. Abhyankar, William Heinzer, and Paul Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310–342. MR 306173, DOI https://doi.org/10.1016/0021-8693%2872%2990134-2
Maria Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Mathematics, vol. 1769, Springer-Verlag, Berlin, 2002. MR 1874328
James W. Alexander, On the factorization of Cremona plane transformations, Trans. Amer. Math. Soc. 17 (1916), no. 3, 295–300. MR 1501043, DOI https://doi.org/10.1090/S0002-9947-1916-1501043-X

[Ca]cast G. Castelnuovo, Le trasformazioni generatrici del gruppo cremoniano nel piano, Atti della Accademia delle Scienze di Torino 36 (1901), 861-874.

A. Crachiola and L. Makar-Limanov, On the rigidity of small domains, J. Algebra 284 (2005), no. 1, 1–12. MR 2115001, DOI https://doi.org/10.1016/j.jalgebra.2004.09.015

[Cr1]cremona1 L. Cremona, Sulle trasformazioni geometriche delle figure piane, Giornale de matematiche di Battaglini 1 (1863), 305-311.

[Cr2]cremona2 L. Cremona, Sulle trasformazioni geometriche delle figure piane, Giornale de matematiche di Battaglini 3 (1865), 269-280, 363-376.

[D]danielewski W. Danielewski, On the cancellation problem and automorphism groups of affine algebraic varieties, preprint, 1989, 8 pages.

M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 523–565, 703 (Russian). MR 0376701
M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. II, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 1, 54–103, 231 (Russian). MR 0437545
Harm Derksen, Ofer Hadas, and Leonid Makar-Limanov, Newton polytopes of invariants of additive group actions, J. Pure Appl. Algebra 156 (2001), no. 2-3, 187–197. MR 1808822, DOI https://doi.org/10.1016/S0022-4049%2899%2900151-6

[Du]dubouloz A. Dubouloz, Generalized Danielewski surfaces, preprint, 2004, 24 pages.

Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619
Karl-Heinz Fieseler, On complex affine surfaces with ${\bf C}^+$-action, Comment. Math. Helv. 69 (1994), no. 1, 5–27. MR 1259603, DOI https://doi.org/10.1007/BF02564471
Gene Freudenburg and Lucy Moser-Jauslin, Embeddings of Danielewski surfaces, Math. Z. 245 (2003), no. 4, 823–834. MR 2020713, DOI https://doi.org/10.1007/s00209-003-0572-5

[HS]hasse H. Hasse and F.K. Schmidt, Noch eine Bergründung der Theorie der höheren Differentialquotienten in einem algebraischen Functionenkörper einer Unbestimmten, J. Reine Angew. Math. 177 (1937), 215-237.

M. Hochster, Nonuniqueness of coefficient rings in a polynomial ring, Proc. Amer. Math. Soc. 34 (1972), 81–82. MR 294325, DOI https://doi.org/10.1090/S0002-9939-1972-0294325-3
I. Martin Isaacs, Algebra, Brooks/Cole Publishing Co., Pacific Grove, CA, 1994. A graduate course. MR 1276273
Heinrich W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161–174 (German). MR 8915, DOI https://doi.org/10.1515/crll.1942.184.161
S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell, $\mathbf C^*$-actions on $\mathbf C^3$ are linearizable, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 63–71. MR 1464577, DOI https://doi.org/10.1090/S1079-6762-97-00025-5
Masanori Koitabashi, Automorphism groups of generic rational surfaces, J. Algebra 116 (1988), no. 1, 130–142. MR 944150, DOI https://doi.org/10.1016/0021-8693%2888%2990196-2
W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953), 33–41. MR 54574
L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math. 69 (1990), no. 2, 250–256. MR 1045377, DOI https://doi.org/10.1007/BF02937308
L. Makar-Limanov, On the group of automorphisms of a surface $x^ny=P(z)$, Israel J. Math. 121 (2001), 113–123. MR 1818396, DOI https://doi.org/10.1007/BF02802499

[N1]noether1 M. Noether, Über Flächen, welche Shaaren rationaler Curven besitzen, Math. Ann. 3 (1871), 161-227.

[N2]noether2 M. Noether, Zur Theorie der eindeutigen Ebenentransformationen, Math. Ann. 5 (1872), 635-639.

[S]segre C. Segre, Un’osservazione relativa alla riducibilità delle trasformazioni cremoniane e dei sistemi lineari di curve piane per mezzo di trasformazioni quadratiche, Atti della Accademia delle Scienze di Torino 36 (1901), 645-651.

Vladimir Shpilrain and Jie-Tai Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), no. 1, 295–307. MR 1900285, DOI https://doi.org/10.1006/jabr.2001.9124
Jörn Wilkens, On the cancellation problem for surfaces, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9, 1111–1116 (English, with English and French summaries). MR 1647227, DOI https://doi.org/10.1016/S0764-4442%2898%2980071-2

Additional Information

Anthony J. Crachiola
Affiliation: Department of Mathematical Sciences, Saginaw Valley State University, 7400 Bay Road, University Center, Michigan 48710
Email: crachiola@member.ams.org

Received by editor(s): November 15, 2004
Published electronically: May 12, 2005