Classification of singular \(\mathbb Q\)-homology planes. II: \(\mathbb C^1\)- and \(\mathbb C^{\ast}\)-rulings (Q453232)

A normal quasi-projective surface \(S\) is called a \(\mathbb{Q}\)-homology plane if \(H_i(S; \mathbb{Q})= (0)\) for \(i>0\). Over the last thirty years many algebraic geometers have made important contributions to the study of these surfaces. This paper is the last step in classifying \(S\) such that the log Kodaira dimension, \(\overline k(S\text{-Sing\,}S)\) of \(S\)-\(\text{Sing\,}S\) is at most 1. Such a surface does not always have a \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration.NEWLINENEWLINE In the first part [``Classification of singular \(\mathbb{Q}\)-homology planes. I. Structure and singularities'', \url{arxiv:0806.3110}], the author had classified \(S\) which have no \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration. In this paper the author assumes that \(S\) has either a \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration. In this case he studies the weighted dual graph in a nice compactification. He studies the singular fibers of the \(\mathbb{C}\) or \(\mathbb{C}^*\)-fibration and gives conditions for \(S\) to be a \(\mathbb{Q}\)-homology plane in terms of the singular fibers. As a corollary of the classification the author describes the number of contractible irreducible algebraic curves on \(S\) in case \(\overline k(S\text{-Sing\,}S)= 0\). This was previously done by \textit{A. J. Parameswaran} and the reviewer [J. Math. Kyoto Univ. 35, No. 1, 63--77 (1995; Zbl 0861.14031)] when \(S\) is smooth.NEWLINENEWLINE The proof uses the theory of open algebraic surfaces developed by Japanese mathematicians, particularly T. Fujita.NEWLINENEWLINE The result in this paper will be valuable to affine algebraic geometers.