Miyanishi's characterization of singularities appearing on \(\mathbb A ^1\)-fibrations does not hold in higher dimensions (Q2850556)

It is known that the existence of a family of rational curves on a variety imposes restrictions on the type of singularities. For example, for \(X\) which is a complex normal affine surface with a \(\mathbb{C}^1\)-fibration the singularities of \(X\) are necessarily cyclic [\textit{T. Kambayashi} and \textit{M. Miyanishi}, Ill. J. Math. 22, 662--671 (1978; Zbl 0406.14012)]. Conversely, any \(2\)-dimensional cyclic quotient singularity can be realized on some \(\mathbb{C}^1\)-fibered affine surface \(X\). Note that, since \(\dim X=2\), the existence of a \(\mathbb{C}^1\)-fibration is equivalent to the existence of an open subset of type \(B\times \mathbb{C}^1\) with \(B\) a smooth curve (a \(\mathbb{C}^1\)-cylinder), see [\textit{M. Miyanishi}, J. Algebra 68, 268--275 (1981; Zbl 0462.14012)]. The following question arises:NEWLINENEWLINE{Question:} Let \(X\) be a normal affine variety which possesses a \(\mathbb{C}^1\)-fibration or a \(\mathbb{C}^1\)-cylinder or a \((\mathbb{C},+)\)-action. What kinds of singularities may appear on \(X\)?NEWLINENEWLINEOne cannot hope that only cyclic quotient singularities appear this way, because (Example 1.1) the restriction of the projection \(\text{Spec} \mathbb{C}[W,Y,U,Z]\to \text{Spec} \mathbb{C}[W,Y]\) to \(X=\{WY=ZU\}\) is a \(\mathbb{C}^1\)-fibration and \(0\in X\) is not a cyclic quotient singularity. Some results in this direction for threefolds have been obtained by the author in [Contemp. Math. 369, 139--163 (2005; Zbl 1074.14052)]. Here the author shows (Theorem 1.2) that any toric (hence any cyclic quotient) singularity can be realized on \(X\) with a \(\mathbb{C}^1\)-fibration. In this case one can additionally assume that the Makar-Limanov invariant of \(\Gamma({\mathcal O}_X)\) is trivial. In fact he shows that on \(\mathbb{C}^1\)-fibered affine varieties one can realize (a more general class of) singularities which admit an effective action of an algebraic torus of some dimension, provided they satisfy some additional combinatorial condition (see Theorem 1.2(2)).NEWLINENEWLINEFor the entire collection see [Zbl 1266.14004].