Transactions of the American Mathematical Society

Author(s): Kayo Masuda; Masayoshi Miyanishi
Journal: Trans. Amer. Math. Soc. 357 (2005), 4867-4883.
MSC (2000): Primary 14R10; Secondary 14R20, 14R25, 14L30
Posted: July 19, 2005
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Abstract: We define affine pseudo-planes as one class of $\mathbb{Q}$ -homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line $\mathbb{A} ^1$ . Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and tom Dieck.

1.: J. Ax, The elementary theory of finite fields, Ann. of Math. 88 (1968), 239-271. MR 0229613 (37:5187)
2.: H. Bass and W. Haboush, Some equivariant K-theory of affine algebraic group actions, Comm. Algebra 15 (1987), 181-217. MR 0876977 (88g:14013)
3.: J. Bertin, Pinceaux de droites et automorphismes des surfaces affines, J. Reine Angew. Math. 341 (1983), 32-53. MR 0697306 (84f:14035)
4.: W. Danielewski, On the cancellation problem and automorphism group of affine algebraic varieties, preprint.
5.: T. tom Dieck, Homology planes without cancellation property, Arch. Math. (Basel) 59 (1992), 105-114. MR 1170634 (93i:14012)
6.: K. H. Fieseler, On complex affine surfaces with $\mathbb{C} ^+$ -action, Comment. Math. Helv. 69 (1994), 5-27. MR 1259603 (95b:14027)
7.: T. Fujita, On the topology of non-complete algebraic surfaces, J. Fac. Sci. of the Univ. Tokyo, Ser. IA 29 (1982), 503-566. MR 0687591 (84g:14035)
8.: R. V. Gurjar and M. Miyanishi, On a generalization of the Jacobian Conjecture, J. Reine Angew. Math. 516 (1999), 115-132. MR 1724617 (2001b:14094)
9.: R.V. Gurjar and M. Miyanishi, Automorphisms of affine surfaces with $\mathbb{A} ^1$ -fibrations, Michigan Math. J. 53 (2005), no. 1, 33-55. MR 2125532
10.: K. Masuda, Certain moduli of algebraic -vector bundles over affine -varieties, in Advanced Studies in Pure Mathematics 33, Math. Soc. of Japan, Tokyo (2002). MR 1890099 (2003a:14096)
11.: K. Masuda and M. Miyanishi, The additive group actions on $\mathbb{Q}$ -homology planes, Ann. Inst. Fourier, Grenoble 53, 2 (2003), 429-464. MR 1990003 (2004e:14073)
12.: K. Masuda and M. Miyanishi, Equivariant cancellation for algebraic varieties, Contemp. Math., 369, Amer. Math. Soc., Providence, RI, 2005. MR 2126662
13.: M. Miyanishi, Lectures on curves on rational and unirational surfaces, Tata Institute of Fundamental Research, Springer, 1978. MR 0546289 (81f:14001)
14.: M. Miyanishi, Open algebraic surfaces, Centre de Recherches Mathématiques, Vol. 12, Université de Montréal, Amer. Math. Soc., 2000. MR 1800276 (2002e:14101)
15.: M. Miyanishi, Singularities of normal affine surfaces containing cylinderlike open sets, J. Algebra 68 (1981), 268-275. MR 0608535 (82k:14035)
16.: M. Miyanishi, Étale endomorphisms of algebraic varieties, Osaka J. Math. 22 (1985), 345-364. MR 0800978 (87f:14021)
17.: M. Miyanishi and K. Masuda, Generalized Jacobian Conjecture and Related Topics, Proceedings of the Internat. Conf. on Algebra, Arithmetic and Geometry, Tata Institute of Fundamental Research, 427-466, 2000. MR 1940676 (2004a:14065)
18.: M. Miyanishi and T. Sugie, Homology planes with quotient singularities, J. Math. Kyoto Univ. 31 (1991), 755-788. MR 1127098 (92g:14034)
19.: G. Schwarz, Exotic algebraic group actions, C. R. Acad. Sci. Paris 309 (1989), 89-94. MR 1004947 (91b:14066)
20.: J.-P. Serre, Espaces fibrés algébriques, Séminaire C. Chevalley, Anneaux de Chow, Exposé 1, 1958.
21.: H. Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1-28. MR 0337963 (49:2732)
22.: J. Wilkens, On the cancellation problem for surfaces, C. R. Acad. Sci. Paris 326 (1998), 1111-1116. MR 1647227 (99i:14040)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14R10, 14R20, 14R25, 14L30

Kayo Masuda
Affiliation: Mathematical Science II, Himeji Institute of Technology, 2167 Shosha, Himeji 671-2201, Japan
Email: kayo@sci.himeji-tech.ac.jp

Masayoshi Miyanishi
Affiliation: School of Science & Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan
Email: miyanisi@ksc.kwansei.ac.jp

DOI: 10.1090/S0002-9947-05-04046-8
PII: S 0002-9947(05)04046-8
Keywords: Equivariant Cancellation Problem, algebraic group action
Received by editor(s): November 26, 2003
Posted: July 19, 2005
Additional Notes: This work was supported by Grant-in-Aid for Scientific Research (C), JSPS
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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