On Higher Order Embeddings of Fano Threefolds by the Anticanonical Linear System

In this article the map given by the anticanonical bundle of a Fano manifold is investigated with respect to a number of natural notions of higher order embeddings of projective manifolds. This is of importance in the understanding of higher order embeddings of the special varieties of adjunction theory, which are usually bered by special Fano manifolds. An analysis is carried out of the higher order embeddings of the special varieties of adjunction theory that arise in the study of the rst and second reductions. Special attention is given to determining the order of the anticanonical embeddings of the three dimensional Fano manifolds which have been classi ed by Iskovskih, Mori, and Mukai and also of the Fano complete intersections in P . 1991 Mathematics Subject Classi cation. Primary 14J45, 14J40; Secondary 14M10, 14N99.