  The main purpose of this paper is to show that OADP varieties stand at an
important crossroad of various main streets in different disciplines like
projective geometry, birational geometry and algebra. This is a good reason for
studying and classifying them. Main specific results are: (a) the
classification of all OADP surfaces (regardless to their smoothness); (b) the
classification of a relevant class of normal OADP varieties of any dimension,
which includes interesting examples like lagrangian grassmannians. Following
[PR], the equivalence of the classification in (b) with the one of
quadro-quadric Cremona transformations and of complex, unitary, cubic Jordan
algebras are explained.
