A note on higher derivations and ordinary points of curves

Abstract

In this note, we prove the following theorem: Let k be an algebraically closed field of arbitrary characteristic.Let C denote a reduced curve in A* and let p be a point of C. Let R denote the local ring of C at p and let R denote the integral closure of R in its total quotient ring.Let M l9 ..., M h be the branches of C at p. Then p is an ordinary point of C if and only if the following two conditions are satisfied: (a) DerJCR) ü Der|(^) for all q ^ 1 ; (b) For t a common uniformizing parameter of C in R, there exists an x in the maximal ideal of R such that dx/dt is a unit in R, and (dx/dt) Mod Mi * (dx/dt) Mod, for all 1 ^ i < j £ h.Introduction.Throughout this paper, we shall let k denote an algebraically closed field of arbitrary characteristic.We shall let C denote a reduced curve in Ag (affine «-space over k).Let p denote a point of C. In this paper, we wish to characterize when p is an ordinary point of C in terms of the higher derivations on the local ring R of C at p. We shall first show that C is unramified at p precisely when every higher order ^-derivation on R extends to the integral closure R. To the best of my knowledge this result was first proven by T. Bloom in [2] for irreducible curves over the complex numbers C.This result was later generalized to arbitrary fields of characteristic zero by J. Becker in [1].In this paper, we present a purely algebraic argument which works in any characteristic.In the last part of this paper, we present straightforward differential conditions which guarantee C has distinct tangents at p. Preliminaries.In this section, we shall present the definitions and basic notation which will be used throughout the rest of this paper.We shall let C denote a reduced curve in AJ.To be more specific; C = Spec{k[Xi, ..., XJ/21} where % is a radical ideal, unmixed of height