Projective geometry of Freudenthal's varieties of certain type (Q1766482)

The purpose of this article is to explore the geometry of Freudenthal-type varieties. To obtain such varieties, one begins with a graded Lie algebra \({\mathfrak g}\) (over \({\mathbb C}\)) of contact type, with \[ {\mathfrak g}=\bigoplus _{-2\leq n\leq 2} {\mathfrak g}_n. \] (One may think of this grading as the natural decomposition of \({\mathfrak g}\) under the adjoint action of a copy of \({\mathfrak {sl}}_2\) within \({\mathfrak g}\).) Then, we define the variety \(V_k\) as the projectivization of \(\{x\in {\mathfrak g_1}\mid (\operatorname{ad} x)^{k+1} {\mathfrak g}_{-2}=0\}\). Aspects of the grading of \({\mathfrak g}\) are important in determining the geometry of the \(V_k\). (For example, the Lie bracket \({\mathfrak g}_1 \times {\mathfrak g}_1\rightarrow {\mathfrak g}_2\) induces a symplectic form on \({\mathfrak g}_1\).) Another important result is the relationship between the quartic form on \({\mathfrak g}_1\) and a certain Cremona transformation associated to \({\mathfrak g}_1\). The authors proceed to establish several geometric results, including that \(V_1\) is a homogeneous space; \(V_1\) is a projectivization of a Lagrangian subvariety of \({\mathfrak g}_1\); \(V_1\) is ruled when \(V_2\neq V\); and numerous results about secants and tangents in the projectivization of \(g_1\).