Transformation Groups, Vol. 11, No. 4, 2006, pp. 659–672 c Birkh¨ auser Boston (2006) AFFINE LINES ON Q-HOMOLOGY PLANES WITH LOGARITHMIC KODAIRA DIMENSION -∞ TAKASHI KISHIMOTO Department of Mathematics Faculty of Science Saitama University Saitama 338-8570, Japan tkishimo@rimath.saitama-u.ac.jp HIDEO KOJIMA Department of Mathematics Faculty of Engineering Niigata University Niigata 950-2181, Japan kojima@ie.niigata-u.ac.jp Abstract. In the present paper, we study a topologically contractible irreducible algebraic curve C on a Q-homology plane S with κ(S)= −∞. We determine such a pair (S, C) when κ(S\C)  0 and C is smooth. Moreover, we prove that if C is not smooth, then C has exactly one singular point and the Makar–Limanov invariant of S is trivial. 0. Introduction Throughout the present paper, we work over the complex number field C. A Q- homology plane is, by definition, a smooth algebraic surface S such that H i (S, Q) = (0) for any positive integer i. Similarly, a Z-homology plane is defined. It is well known that any Q-homology plane is affine and rational (cf. [7], [17]). Let C ⊂ C 2 be a topologically contractible irreducible algebraic curve. The Abhyan- kar–Moh–Suzuki theorem (cf. [1] and [33]) (resp., the Lin–Zaidenberg theorem (cf. [26])) says that if C is smooth (resp., C is not smooth), then there exists a system of coordi- nates {X,Y } on C 2 such that the curve C is defined by {X =0} (resp., {X m = Y n }, where 0 <m<n and gcd(m,n) = 1). In particular, if C is smooth (resp., C is not smooth), then κ(C 2 \ C)= −∞ (resp., κ(C 2 \ C) = 1 and C has exactly one singular point). Later on, several authors studied topologically contractible algebraic curves on Q-homology planes. Zaidenberg [35] proved that a Z-homology plane of logarithmic Ko- daira dimension 2 (resp., a Z-homology plane  ∼ = C 2 of logarithmic Kodaira dimension  1) contains no topologically contractible algebraic curves (resp., a unique topologi- cally contractible algebraic curve; it is actually smooth). Miyanishi and Tsunoda [31], Gurjar and Miyanishi [14] and Gurjar and Parameswaran [18] studied topologically con- tractible algebraic curves on Q-homology planes with nonnegative logarithmic Kodaira dimension. More precisely, the following results are known. Theorem 0.1. Let S be a Q-homology plane with κ(S)  0. Then the following asser- tions hold: (1) (cf. [31], [14]). If κ(S)=2, then S contains no topologically contractible algebraic curves. DOI: 10.1007/S00031-005-1121-6. Received August 16, 2005. Accepted March 9, 2006.