Classification of singular ℚ-homology planes, II: ℂ1- and ℂ∗-rulings

A ‫-ޑ‬homology plane is a normal complex algebraic surface having trivial rational homology.We classify singular ‫-ޑ‬homology planes that are ‫ރ‬ 1or ‫ރ‬ * -ruled.We analyze their completions, the number of different rulings they have, and the number of affine lines on them; and we give constructions.Together with previously known results, this completes the classification of ‫-ޑ‬homology planes with smooth locus of nongeneral type.We show also that the dimension of a family of homeomorphic but nonisomorphic singular ‫-ޑ‬homology planes having the same weighted boundary, singularities and Kodaira dimension can be arbitrarily big.We work with complex algebraic varieties. Main resultsA ‫-ޑ‬homology plane is a normal surface whose rational cohomology is the same as that of ‫ރ‬ 2 .This paper is the last piece of the classification of ‫-ޑ‬homology planes having smooth locus of nongeneral type.The classification is built on the work of many authors; for a summary of what is known about smooth and singular ‫-ޑ‬ homology planes, see [Miyanishi 2001, §3.4] and [Palka 2011b].In [Palka 2008], we classified singular ‫-ޑ‬homology planes with nonquotient singularities, showing in particular that they are quotients of affine cones over projective curves by actions of finite groups that respect the set of lines through the vertex.In [Palka 2011a], we classified singular ‫-ޑ‬homology planes whose smooth locus is of nongeneral type and admits no ‫ރ‬ 1 -or ‫ރ‬ * -ruling (exceptional planes).Here we classify singular ‫-ޑ‬homology planes that admit a ‫ރ‬ 1 -or a ‫ރ‬ * -ruling.We analyze completions and boundaries rather than the open surfaces themselves.To deal with nonuniqueness of these, we use the notions of a balanced and a strongly balanced weighted boundary and completion of an open surface (see Definitions 2.7 and 2.10).We classify ‫ރ‬ 1 -and ‫ރ‬ * -ruled ‫-ޑ‬homology planes by giving necessary and sufficient conditions for a ‫ރ‬ 1 -or ‫ރ‬ * -ruled open surface to be a ‫-ޑ‬homology plane The author was supported by Polish Grant NCN N N201 608640.