  The Severi and Scorza varieties are the limiting cases of a theorem of Zak
conjectured by Hartshorne. Zak also classified the Severi and Scorza varieties.
Surprisingly enough, there are only 4 Severi varieties, one for each dimension
2,4,8 and 16 and they are homogeneous and very strongly linked with the 4
rank-3 Jordan algebras. In this article, I give several variations of Zak's
classification theorem, proving a priori the homogeneity of Scorza varieties,
and showing how it is possible to give a Jordan structure to the ambient space
of a Scorza variety.
