Affine lines on logarithmic \(Q\)-homology planes (Q1195997)

The following result (in a slightly general form) is proved: Theorem: Let \(X\) be a normal affine surface\(/\mathbb{C}\) having at most quotient singularities and with \(H_ i(X;\mathbb{Z})=(0)\) for \(i\geq 1\). Then (1) If \(\bar k(X)=2\), then \(X\) contains no algebraic curve homeomorphic to \(\mathbb{A}^ 1\). (2) If \(\bar k(X)=1\), then \(X\) contains a unique algebraic curve homeomorphic to \(\mathbb{A}^ 1\). It is actually isomorphic to \(\mathbb{A}^ 1\) and contains at most one (cyclic) quotient singular point of \(X\). This result in the smooth case was proved by Miyanishi, Tsunoda and Sugie. Using recent work of Peter Russell, M. Koras, Hanspeter Kraft and the authors on the linearization problem of a \(\mathbb{C}^*\) action on \(\mathbb{C}^ 3\), from the above theorem it follows that if \(\mathbb{C}^ 3//\mathbb{C}^*\) contains a contractible algebraic curve, then the \(\mathbb{C}^*\) action is linearizable.