On Quasihomogeneous Curves

A hypersurface is said to be quasihomogeneous if in suitable coordinates with assigned weights, its equation becomes weighted homogeneous in its variables. For an irreducible quasihomogeneous plane curve, the equation necessarily becomes a two term equation of the form aY n + bX where n,m are necessarily coprime. Zariski, in a short paper, established a criterion for an algebroid curve to be quasihomogeneous [Z1] and a celebrated theorem of Lin and Zaidenberg gives a global criterion for quasihomogeneity [LZ]. The Lin-Zaidenberg theorem does not have a simple proof, despite having three different proofs using function theory, topology and algebraic surface theory respectively. We give here a global version of the Zariski result. As a consequence we give a proof of a slightly weaker version of the Lin-Zaidenberg Theorem, namely that a rational curve with one place at infinity is unibranch and locally quasihomogeneous if and only if it is globally quasihomogeneous, provided the ground field is algebraically closed of characteristic zero. Our method of proof leads to some interesting questions about the change in the module of differentials when we go to the integral closure.