Globalization of an old Theorem of Zariski
by
Avinash Sathaye
University of Kentucky, Lexington KY 40506 U.S.A.
Coauthors: Abdallah Assi, Université d'Angers, Mathématiques, 49045 Angers cedex 01, France

I will describe the joint work with Abdallah Assi (On Quasihomogeneous Curves - preprint). Let the ground field be algebraically closed of characteristic zero. In a short paper (1966: Collected Works V.3, p. 475-480), Zariski characterized plane unibranch curves having maximum torsion to be exactly curves of the form y^a-x^b with a, b coprime, after a suitable local change of variables. This torsion came out to be the length of the module of differentials of the integral closure modulo the module of differentials of the original coordinate ring (W([`R])/W(R)). We show that this concept can be defined similarly for an affine curve with one place at infinity and prove a similar characterization, namely the length is maximal if and only if the curve is of Lin-Zaidenberg type, meaning after a suitable affine change of coordinates, it is of the form y^a-x^b with a, b coprime. The well known Abhyankar Moh theory of plane affine curves with one place at infinity gives necessary conditions for such a curve to be rational, but the characterization of such curves has not been known. The relative module of differentials gives a new tool to study additional conditions on a plane curve with one place at infinity imposed by its rationality. We will describe additional calculations of this relative module of differentials.

Date received: March 30, 2005