On the Projective Embedding of Homogeneous Varieties

In an article [5] elsewhere in this volume, Weil has shown that an idea of Lefschetz on the projective embedding of an Abelian variety over the complex field, which seemingly depends upon the use of theta functions ([1], pp. 368–9), can actually be extended to the case of an abstract Abelian variety over a field of arbitrary characteristic. In this note we shall show that this idea can be further extended to get a projective embedding not only of an arbitrary group variety, but also of any homogeneous variety. We shall say that a variety V can be embedded in a projective space or has a projective embedding, if there is an everywhere biregular birational transformation of V onto a (not necessarily complete) variety contained in a projective space…