Affine Lines on Affine Surfaces and the Makar–Limanov Invariant

Gurjar, R. V. ; Masuda, K. ; Miyanishi, M. ; Russell, P.

Canadian journal of mathematics, Tome 60 (2008) no. 1, pp. 109-139

Résumé

A smooth affine surface $X$ defined over the complex field $\mathbb{C}$ is an $\text{M}{{\text{L}}_{0}}$ surface if the Makar–Limanov invariant $\text{ML(}X\text{)}$ is trivial. In this paper we study the topology and geometry of $\text{M}{{\text{L}}_{0}}$ 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 ${{\mathbb{A}}^{1}}$ -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 $\text{M}{{\text{L}}_{0}}$ property under proper maps.

Détail

DOI : 10.4153/CJM-2008-005-8

Mots-clés : Primary, 14R20, secondary, 14L30

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