Logarithmic plurigenera of smooth affine surfaces with finite Picard groups

Abstract

Let S be a smooth complex affine surface with finite Picard group. We prove that if ¯ κ(S) = 1 (resp. ¯ κ(S) = 2), then ¯ P2(S) > 0 (resp. ¯ P6(S) > 0) and determine the surface S when ¯ κ(S) ≥ 0 and ¯ P6(S) = 0. Moreover, we prove that if Pic(S) = (0), �(S, OS) ∗ = C ∗ and ¯ P2(S) = 0, then S ∼ C 2 .