Gurjar, R. V.; Masuda, K.; Miyanishi, M.; Russell, P. Affine lines on affine surfaces and the Makar-Limanov invariant. (English) Zbl 1137.14049 Can. J. Math. 60, No. 1, 109-139 (2008). The authors consider smooth affine surfaces admitting \(\mathbb C^+\)-fibrations. A \(\mathbb C^+\)-fibration is a morphism \(f:X\to \Gamma\) into a curve \(\Gamma,\) such that for a general point \(\gamma\in \Gamma\) the fiber \(F_{\gamma}=f^{-1}(\gamma)\cong \mathbb C.\) These fibrations are connected to the Makar-Limanov invariant \(ML(X).\) Namely:(1) If a surface \(X\) admits \(\mathbb C^+\)-fibrations to affine curves and all that fibrations have the same general fiber, then \(ML(X)=\mathbb C[\Gamma].\) The authors call such surfaces \(ML_1;\)(2) If there are at least two \(\mathbb C^+\)-fibrations from \(X\) to affine curves with different general fibers, then \(ML(X)=\mathbb C.\) In this case \(X\) is called \(ML_0.\)A curve \(C\subset X\) is called anomalous if \(C\cong \mathbb C\) but is not a connected component of a fiber of some \(\mathbb C^+\)-fibration.The paper provides an answer to the following question.Is it possible that a smooth affine surface \(X\) admitting \(\mathbb C^+\)-fibration contains an anomalous curve?The answer is positive in general. An example of an \(ML_0\) surface \(X\) with \(\rho (X)=\text{rank} ( \text{Pic}(X)_{\mathbb Q})=1,\) which contains an anomalous curve, is given in Theorem 2.3. Two examples of \(ML_1\) surfaces \(X_1,X_2,\) containing an anomalous curve, are also provided.The answer is proven to be negative in the following cases. 1.\(X\) is \(ML_0\) and \(\rho (X)=0;\) 2.\(X\) is \(ML_1\) and \(\mathbb Q\)-homology plane and \(X\) is not isomorphic to \(X_1\) or \(X_2\) (see above); 3.\(X\) admits a \(\mathbb C^+\)-fibration \(f:X\to \Gamma, \) such that \(\bullet\)\(\Gamma \) is smooth and isomorphic to \(\mathbb P^1\) or \(\mathbb C\); \(\bullet\)every fiber is irreducible; \(\bullet\)\(f\) has at least two (resp. 3) multiple fibers if \(\Gamma \cong\mathbb C\) (resp. \(\Gamma \cong\mathbb P^1\)). Another question considered in the paper is whether the property being \(ML_i, i=1,2,\) is preserved under the étale finite morphisms.The following Theorem is proved.Let \(\phi :X\to Y\) be an étale morphism. Then \(Y\) is \(ML_i, i=1,2,\) if and only if \(X\) is also. Reviewer: Tatiana Bandman (Ramat-Gan) Cited in 5 ReviewsCited in 16 Documents MSC: 14R25 Affine fibrations 14R20 Group actions on affine varieties 14L30 Group actions on varieties or schemes (quotients) Keywords:affine surface; \(\mathbb{C}\)-fibration; rational curve; Makar-Limanov invariant; finite morphism PDFBibTeX XMLCite \textit{R. V. Gurjar} et al., Can. J. Math. 60, No. 1, 109--139 (2008; Zbl 1137.14049) Full Text: DOI