On automorphisms of Danielewski surfaces
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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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.
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[AEH]AEH S. Abhyankar, P. Eakin, and W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310-342.
[Alb]alberich M. Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Math., Vol. 1769, Springer, Berlin, 2002.
[Ale]alexander J.W. Alexander, On the factorization of Cremona plane transformations, Trans. Amer. Math. Soc. 17 (1916), 295-300.
[Ca]cast G. Castelnuovo, Le trasformazioni generatrici del gruppo cremoniano nel piano, Atti della Accademia delle Scienze di Torino 36 (1901), 861-874.
[CM]CML A. Crachiola and L. Makar-Limanov, On the rigidity of small domains, J. Algebra 284 (2005), no. 1, 1–12.
[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.
[DG1]dg1 V.I. Danilov and M.H. Gizatulin, Automorphisms of affine surfaces I, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 523-565. English translation: Math. USSR Izv. 9 (1975), 493-534.
[DG2]dg2 V.I. Danilov and M.H. Gizatulin, Automorphisms of affine surfaces II, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 54-103. English translation: Math. USSR Izv. 11 (1977), 51-98.
[DHM]DHML H. Derksen, O. Hadas, and L. Makar-Limanov, Newton polytopes of invariants of additive group actions, J. Pure Appl. Algebra 156 (2001), 187-197.
[Du]dubouloz A. Dubouloz, Generalized Danielewski surfaces, preprint, 2004, 24 pages.
[E]essen A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progr. Math., Vol. 190, Birkhäuser Verlag, Basel, 2000.
[F]fieseler K.-H. Fieseler, On complex affine surfaces with $\mathbf {C}^+$-action, Comment. Math. Helv. 69 (1994), 5-27.
[FM]freudenburg G. Freudenburg and L. Moser-Jauslin, Embeddings of Danielewski surfaces, Math. Z. 245 (2003), 823-834.
[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.
[H]hochster M. Hochster, Nonuniqueness of coefficient rings in a polynomial ring, Proc. Amer. Math. Soc. 34 (1972), 81-82.
[I]isaacs I.M. Isaacs, Algebra, a graduate course, Brooks/Cole, Pacific Grove, California, 1994.
[J]jung H.W.E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161-174.
[KKMR]KKMLR 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.
[Ko]koitabashi M. Koitabashi, Automorphism groups of generic rational surfaces, J. Algebra 116 (1988), 130-142.
[K]kulk W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde (3) 1 (1953), 33-41.
[M1]ML1 L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math. 69 (1990), 250-256.
[M2]ML2 L. Makar-Limanov, On the group of automorphisms of a surface $x^n y = P(z)$, Israel J. Math. 121 (2001), 113-123.
[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.
[SY]shpilrain V. Shpilrain and J.-T. Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), no. 1, 295-307.
[W]wilkens J. Wilkens, On the cancellation problem for surfaces, C.R. Acad. Sci. Paris Sér. I 326 (1998), 1111-1116.
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