The secant varieties of nilpotent orbits
Let \(\mathfrak g\) be a complex simple Lie algebra endowed with the adjoint action of the adjoint group \(G\). Let \({\mathbf P}{\mathfrak g}\) be the projective space of one-dimensional linear subspaces in \(\mathfrak g\) and let \(\pi: \mathfrak g\setminus \{0\}\to {\mathbf P}{\mathfrak g}\) be the natural projection. The classification of nonzero nilpotent \(G\)-orbits in \(\mathfrak g\) is well known. Let \(\mathcal O\) be such an orbit and let \(X\) be the closure of \(\pi(\mathcal O)\) in \({\mathbf P}{\mathfrak g}\). Denote by \({\text{Sec}}^{(i)}X\) the \(i\)-secant variety of \(X\), i.e., the closure of the union of projective subspaces of \({\mathbf P}{\mathfrak g}\) of dimension \(i\) spanned by \(i+1\) points of \(X\). The author explicitly describes \({\text{Sec}}^{(i)}X\) for all classical simple Lie algebras \(\mathfrak g\) and all nonzero nilpotent orbits \(\mathcal O\). For instance, if \(\mathfrak g=\mathfrak{s}\mathfrak{l}_n\) and \(\text{rk}(x)=k\) for \(x\in \mathcal O\), then \({\text{Sec}}^{(i)}X=\{y\in \mathfrak{s}\mathfrak{l}_n\mid \text{rk}(y)\leq k(i+1)\}\) for \(i\geq 2\).
scientific article; zbMATH DE number 5363918
scientific article