location:  Publications → journals → CJM
Abstract view

# 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
 Format: HTML LaTeX MathJax PDF PostScript

## 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]