The additive group actions on ${\mathbb {Q}}$-homology planes

Masuda, Kayo; Miyanishi, Masayoshi

doi:10.5802/aif.1949

The additive group actions on

ℚ

-homology planes
[Actions du groupe additif sur des plans

ℚ

-acycliques]

Masuda, Kayo ¹ ; Miyanishi, Masayoshi

¹ Himeji Institute of Technology, Mathematical Sciences II, 2167 Shosha, Himeji 671-2201 (Japon)

Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 429-464.

Résumé
Abstract

Dans cet article, on démontre qu’un plan $ℚ$ -acyclique $X$ avec deux actions du groupe additif $G_{a}$ qui sont algébriquement indépendantes, est isomorphe au plan affine ou bien au quotient d’une hypersurface affine $x y = z^{n} - 1$ dans l’espace affine de dimension 3 par une action de $ℤ / m ℤ$ , où $m$ est l’ordre d’un groupe fini $H_{1} (X; ℤ)$

In this article, we prove that a $ℚ$ -homology plane $X$ with two algebraically independent $G_{a}$ -actions is isomorphic to either the affine plane or a quotient of an affine hypersurface $x y = z^{m} - 1$ in the affine $3$ -space via a free $ℤ / m ℤ$ -action, where $m$ is the order of a finite group $H_{1} (X; ℤ)$ .

MR Zbl

DOI : 10.5802/aif.1949

Classification : 14L30, 14R20, 14J26
Keywords: ${\mathbb {Q}}$-homology plane, additive group action, Makar-Limanov invariant
Mot clés : plan ${\mathbb {Q}}$-acyclique, action du groupe additif, invariant de Makar-Limanov

Affiliations des auteurs :

Masuda, Kayo ¹ ; Miyanishi, Masayoshi

¹ Himeji Institute of Technology, Mathematical Sciences II, 2167 Shosha, Himeji 671-2201 (Japon)

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     title = {The additive group actions on ${\mathbb {Q}}$-homology planes},
     journal = {Annales de l'Institut Fourier},
     pages = {429--464},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     doi = {10.5802/aif.1949},
     mrnumber = {1990003},
     zbl = {1085.14054},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1949/}
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Masuda, Kayo; Miyanishi, Masayoshi. The additive group actions on ${\mathbb {Q}}$-homology planes. Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 429-464. doi : 10.5802/aif.1949. http://www.numdam.org/articles/10.5802/aif.1949/

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