Logarithmic plurigenera of some affine surfaces

I will present recent results on logarithmic plurigenera of smooth affine
surfaces with finite Picard groups. The main result says that, for a smooth
affine surface S with finite Picard group, S has non-negative logarithmic
Kodaira dimension if and only if it has positive logarithmic 6 genus.
Moreover, I will give more precise results when S is a complement of a
plane curve.