  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 fibered 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 first and second reductions. Special attention is
given to determining the order of the anticanonical embeddings of the three
dimensional Fano manifolds which have been classified by Iskovskih, Mori, and
Mukai and also of the Fano complete intersections in $\pn N$.

Algebraic Geometry;Algebraic Geometry