Some projective varieties defined by Jordan algebras
Let J be a finite-dimensional (unital quadratic) Jordan algebra over a field F. An element \(c\in J\) is called reduced if \(JU_ c\subseteq Fc\); an algebra J is called reduced if it is spanned by reduced elements. Let \(\bar F\) be an algebraically closed field containing F. The author defines the (projective) variety \(V_ J\) of reduced elements as the set of points \(\bar Fv\) in the projective space \({\mathbb{P}}J_{\bar F}\) of one- dimensional subspaces of \(J_{\bar F}\) such that \(v\neq 0\) is a reduced element of \(J_{\bar F}\). This is a variety defined by homogeneous polynomials with coefficients in F. It is shown that if J is outer central simple, then \(V_ J\) is absolutely irreducible and, moreover, \(V_{J_{F_ s}}\) for \(F_ s\) the separable algebraic closure of F is birationally equivalent to a projective space. Further, the author calls an extension field E/F to be a reducing field for J if \(J_ E\) is reduced; E is called a generic reducing field if it has the property that an extension E'/F is a reducing field for J if and only if there exists an F-place on E to E'. The irreducibility of \(V_ J\) for J outer central simple permits the author to define the field \(PF(V_ J)\) of rational functions on \(V_ J\). It is shown that this field is a generic reducing field for J, and in this case J is reduced if and only if \(PF(V_ J)\) is purely transcendental over F. Another projective variety associated with a Jordan algebra J is a norm hypersurface \(N_ J\) of J, which the author defines as the set of zeros in \({\mathbb{P}}J_{\bar F}\) of the generic norm N(x). If J is outer central simple, then \(N_ J\) is absolutely irreducible, the field of rational functions (homogeneous of degree 0) on \(N_ J\) is a generic reducing field for J and J is reduced if and only if this field is purely transcendental over F. See also the review of \textit{H. P. Petersson}'s article ''Generic reducing fields of Jordan pairs'', Trans. Am. Math. Soc. 285, 825-843 (1984) in Zbl 0575.17010.
scientific article; zbMATH DE number 3931222
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