We study some properties of the affine plane. First we describe the set of fixed points of a polynomial automorphism of ℂ². Next we classify completely so-called identity sets for polynomial automorphisms of ℂ². Finally, we show that a sufficiently general Zariski open affine subset of the affine plane has a finite group of automorphisms.
@article{bwmeta1.element.bwnjournal-article-apmv60z2p163bwm,
author = {Zbigniew Jelonek},
title = {The automorphism groups of Zariski open affine subsets of the affine plane},
journal = {Annales Polonici Mathematici},
volume = {60},
year = {1994},
pages = {163-171},
zbl = {0824.14011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv60z2p163bwm}
}
Zbigniew Jelonek. The automorphism groups of Zariski open affine subsets of the affine plane. Annales Polonici Mathematici, Tome 60 (1994) pp. 163-171. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv60z2p163bwm/
[000] [A-M] S. Abhyankar and T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166. | Zbl 0332.14004
[001] [Iit1] S. Iitaka, On logarithmic Kodaira dimension of algebraic varieties, in: Complex Analysis and Algebraic Geometry, Iwanami, Tokyo, 1977, 178-189.
[002] [Iit2] S. Iitaka, Birational Geometry for Open Varieties, Les Presses de L'Université de Montréal, 1981.
[003] [Jel1] Z. Jelonek, Identity sets for polynomial automorphisms, J. Pure Appl. Algebra 76 (1991), 333-339. | Zbl 0752.14010
[004] [Jel2] Z. Jelonek, Irreducible identity sets for polynomial automorphisms, Math. Z. 212 (1993), 601-617. | Zbl 0806.14011
[005] [Jel3] Z. Jelonek, Affine smooth varieties with finite group of automorphisms, ibid., to appear.
[006] [Jel4] Z. Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), 113-120. | Zbl 0611.14010
[007] [Jel5] Z. Jelonek, Sets determining polynomial automorphisms of ℂ², Bull. Polish Acad. Sci. Math. 37 (1989), 247-250.
[008] [Kal] S. Kaliman, Polynomials on ℂ² with isomorphic generic fibers, Soviet Math. Dokl. 33 (1986), 600-603. | Zbl 0623.14007
[009] [Kam] T. Kambayashi, Automorphism group of a polynomial ring and algebraic group action on an affine space, J. Algebra 60 (1979), 439-451. | Zbl 0429.14017
[010] [Kul] W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. 1 (1953), 33-41. | Zbl 0050.26002
[011] [M-W] J. MacKay and S. Wang, An inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 76 (1986), 245-257. | Zbl 0622.13003
[012] [Sak] F. Sakai, Kodaira dimension of complements of divisors, in: Complex Analysis and Algebraic Geometry, Iwanami, Tokyo, 1977, 239-257.
[013] [Suz1] M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace ℂ², J. Math. Soc. Japan 26 (1974), 241-257. | Zbl 0276.14001
[014] [Suz2] M. Suzuki, Sur les opérations holomorphes du groupe additif complexe sur l'espace de deux variables complexes, Ann. Sci. Ecole Norm. Sup. 10 (1977), 517-546. | Zbl 0403.32020
[015] [Zai] M. G. Zaĭdenberg, Isotrivial families of curves on affine surfaces and characterization of the affine plane, Math. USSR-Izv. 30 (1988), 503-532. | Zbl 0666.14018
[016] [Z-L] M. G. Zaĭdenberg and V. Ya. Lin, An irreducible simply connected algebraic curve in ℂ² is equivalent to a quasihomogeneous curve, Soviet Math. Dokl. 28 (1983), 200-203.