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Affine Lines on Affine Surfaces and the Makar--Limanov Invariant |
Canad. J. Math. 60(2008), 109-139
Published:2008-02-01
Printed: Feb 2008
Printed: Feb 2008
- R. V. Gurjar
- K. Masuda
- M. Miyanishi
- P. Russell
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 - Group actions on affine varieties [See also 13A50, 14L30] 14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] |