  Using the theory of minimal models of quasi-projective surfaces we give a new
proof of the theorem of Lin-Zaidenberg which says that every topologically
contractible algebraic curve in the complex affine plane has equation $X^n=Y^m$
in some algebraic coordinates on the plane. This gives also a proof of the
theorem of Abhyankar-Moh-Suzuki concerning embeddings of the complex line into
the plane. Independently, we show how to deduce the latter theorem from basic
properties of $\mathbb{Q}$-acyclic surfaces.
