  This is a survey of our research on geometric structures of projective
embeddings and includes some topics of our talks in several symposia during
1990-99. We clarify our main problem, which is to construct a kind of geometric
composition series of projective embeddings. The concept of "geometric
composition series" is an analogy in Algebraic Geometry with Jordan-H\"{o}lder
series in Group Theory. We present two of the candidates for the construction
problem. To approach this problem, we show several results and new tools for
handling higher obstructions appearing in infinitesimal liftings. As a
byproduct of the tools, we obtain a simplified proof for a criterion on
arithmetic normality described in terms of Differential Geometry. (Keywords):
Petri's analysis, Second fundamental form, Syzygy, Arithmetic normality,
Infinitesimal lifting, Geometric shell, Geometric composition series, Lefschetz
chain, Dual Lefschetz chain, Meta-Lefschetz operator.
