Singularities of Projective Embedding (Points of order n on an Elliptic Curve)

Abstract

In the Plücker formula for a curve embedded in a higher dimensional projective space, one encounters the notion of stationary point (cf, [B], [W]). W. F. Pohl gave new view point about it in terms of vector bundles and he defined “the singularities of embedding” (cf. [P]). At first, we shall give dual formulation of Pohl’s one by means of the sheaf of principal parts of order n, and next we shall prove the following: If an elliptic curve is embedded in (n — l)-dimensional projective space as a curve of degree n, singularities of projective embedding of order n — 1 are exactly the points of order n with suitable choice of a neutral element on the curve which is an abelian variety of dimension one. The proof is given by making use of the relation between and Schwarzenberger’s secant bundle which we shall also give.