On the classification of OADP varieties

TL;DR

This paper classifies OADP varieties, revealing their significance across multiple disciplines like projective and birational geometry, and connects their classification to Cremona transformations and Jordan algebras.

Contribution

It provides the first comprehensive classification of all OADP surfaces and a relevant class of higher-dimensional OADP varieties, linking them to Cremona transformations and Jordan algebras.

Findings

Classified all OADP surfaces regardless of smoothness.

Classified a key class of higher-dimensional normal OADP varieties.

Established connections between OADP varieties, Cremona transformations, and Jordan algebras.

Abstract

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.