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Affine lines on affine surfaces and the Makar-Limanov invariant. (English) Zbl 1137.14049

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.

MSC:

14R25 Affine fibrations
14R20 Group actions on affine varieties
14L30 Group actions on varieties or schemes (quotients)
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