Scorza varieties and Jordan algebras (Q1433042)

[\textit{F. L. Zak}, Tangents and secants of algebraic varieties. Translations of Mathematical Monographs. 127. Providence, RI: American Mathematical Society. (1993; Zbl 0795.14018)] introduced the notion of a Scorza variety and discovered their relation with Jordan algebras. Given a fixed number \(k\), a \textit{Scorza variety} is a smooth projective variety which is of maximal dimension among all such varieties for which the \((k-1)\)-secant variety is not all of the ambient space. A \textsl{Jordan algebra} is a commutative algebra which is not necessarily associative, but fulfills the weaker law \(A(BA^2)=(AB)A^2\). In this article the author gives new proofs for the correspondence between these two concepts. They provide some new insights, in particular concerning the homogeneity of Scorza varieties and automorphism groups of Jordan algebras.