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.
Categories: math.AG