The additive group actions on ℚ -homology planes

Kayo Masuda[1]; Masayoshi Miyanishi

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

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 2, page 429-464
  • ISSN: 0373-0956

Abstract

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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 ) .

How to cite

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Masuda, Kayo, and Miyanishi, Masayoshi. "The additive group actions on ${\mathbb {Q}}$-homology planes." Annales de l’institut Fourier 53.2 (2003): 429-464. <http://eudml.org/doc/116042>.

@article{Masuda2003,
abstract = {In this article, we prove that a $\{\mathbb \{Q\}\}$-homology plane $X$ with two algebraically independent $G_a$-actions is isomorphic to either the affine plane or a quotient of an affine hypersurface $xy=z^m-1$ in the affine $3$-space via a free $\{\mathbb \{Z\}\}/m\{\mathbb \{Z\}\}$-action, where $m$ is the order of a finite group $H_1(X;\{\mathbb \{Z\}\})$.},
affiliation = {Himeji Institute of Technology, Mathematical Sciences II, 2167 Shosha, Himeji 671-2201 (Japon)},
author = {Masuda, Kayo, Miyanishi, Masayoshi},
journal = {Annales de l’institut Fourier},
keywords = {$\{\mathbb \{Q\}\}$-homology plane; additive group action; Makar-Limanov invariant; -homology plane; additive group actions},
language = {eng},
number = {2},
pages = {429-464},
publisher = {Association des Annales de l'Institut Fourier},
title = {The additive group actions on $\{\mathbb \{Q\}\}$-homology planes},
url = {http://eudml.org/doc/116042},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Masuda, Kayo
AU - Miyanishi, Masayoshi
TI - The additive group actions on ${\mathbb {Q}}$-homology planes
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 2
SP - 429
EP - 464
AB - In this article, we prove that a ${\mathbb {Q}}$-homology plane $X$ with two algebraically independent $G_a$-actions is isomorphic to either the affine plane or a quotient of an affine hypersurface $xy=z^m-1$ in the affine $3$-space via a free ${\mathbb {Z}}/m{\mathbb {Z}}$-action, where $m$ is the order of a finite group $H_1(X;{\mathbb {Z}})$.
LA - eng
KW - ${\mathbb {Q}}$-homology plane; additive group action; Makar-Limanov invariant; -homology plane; additive group actions
UR - http://eudml.org/doc/116042
ER -

References

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  7. R.V. Gurjar, C.R. Pradeep, Q -homology planes are rational. III, Osaka J. Math 36 (1999), 259-335 Zbl0954.14013MR1736480
  8. S. Kaliman, L. Makar-Limanov, On the Russell-Koras contractible threefolds. J. Algebraic Geom, J. Algebraic Geom 6 (1997), 247-268 Zbl0897.14010MR1489115
  9. K. Masuda, M. Miyanishi, Étale endomorphisms of algebraic surfaces with G m -actions, Math. Ann 319 (2001), 493-516 Zbl1016.14028MR1819880
  10. M. Miyanishi, Curves on rational and unirational surfaces, (1978), Springer Zbl0425.14008MR546289
  11. M. Miyanishi, K. Masuda, Generalized Jacobian conjecture and related topics, Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry, Mumbai 2000, Tata Institute of Fundamental Research (2002), Narosa Zbl1053.14071
  12. M. Miyanishi, T. Sugie, Homology planes with quotient singularities, J. Math. Kyoto Univ 31 (1991), 755-788 Zbl0790.14034MR1127098

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