We verify a conjecture of Miyanishi about smooth quasi-projective surfaces over the complex
numbers, namely, either such a surface has a pluricanonical log form, or a general point of the
surface lies on a one dimensional image of the affine line. Applying standard techniques of the
minimal model program, we reduce the problem to a detailed study of log del Pezzo surfaces.
Log del Pezzo surfaces are normal projective surfaces, with ample anticanonical divisor and
quotient singularities. WTe also give a partial classification of log del Pezzo surfaces of Picard
1991 Mathematics Subject Classification. 14J26 and 14J45.
Key words and phrases. Log del Pezzo surface, log uniruled, quasi-projective surfaces, higher dimensional
geometry, minimal model program, rational curves.