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 $\bar{\kappa }(S)=1$ (resp. $\bar{\kappa }(S)=2$) then ${\bar{P}}_2(S)>0$ (resp. ${\bar{P}}_6(S)>0$) and determine the surface $S$ when $\bar{\kappa }(S)\ge 0$ and ${\bar{P}}_6(S)=0$. Moreover, we prove that if $\mathrm{Pic}(S)=(0)$, $\Gamma (S,{\mathcal{O}}_S{)}^{\ast }=C^{\ast }$ and ${\bar{P}}_2(S)=0$ then $S\cong C^2$.