I will discuss an ongoing project (joint with H. Van Maldeghem) to give a uniform axiomatic description of the embeddings in projective space of the varieties corresponding with the geometries of exceptional Lie type over arbitrary fields.
In particular, I will provide a uniform geometric characterization of the Severi varieties over arbitrary fields, i.e. the quadric Veronese varieties in 5- dimensional projective spaces, the Segre varieties in 8-dimensional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the 26-dimensional exceptional varieties of type E6
