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Affine Lines on Affine Surfaces and the Makar--Limanov Invariant

  Published:2008-02-01
 Printed: Feb 2008
  • R. V. Gurjar
  • K. Masuda
  • M. Miyanishi
  • P. Russell
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Abstract

A smooth affine surface $X$ defined over the complex field $\C$ is an $\ML_0$ surface if the Makar--Limanov invariant $\ML(X)$ is trivial. In this paper we study the topology and geometry of $\ML_0$ surfaces. Of particular interest is the question: Is every curve $C$ in $X$ which is isomorphic to the affine line a fiber component of an $\A^1$-fibration on $X$? We shall show that the answer is affirmative if the Picard number $\rho(X)=0$, but negative in case $\rho(X) \ge 1$. We shall also study the ascent and descent of the $\ML_0$ property under proper maps.
MSC Classifications: 14R20, 14L30 show english descriptions Group actions on affine varieties [See also 13A50, 14L30]
Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
14R20 - Group actions on affine varieties [See also 13A50, 14L30]
14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
 

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