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) , Γ(S,\mathcal{O}S)* = ℂ* and P2(S) = 0 then S ≅ ℂ2.