$${{\,\mathrm{\mathbb {P}GL}\,}}$$ Orbits in Tree Varieties

Abstract In this paper, we introduce tree varieties as a natural generalization of products of partial flag varieties. We study orbits of the $${{\,\mathrm{\mathbb {P}GL}\,}}$$ action on tree varieties. We characterize tree varieties with finitely many $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbits, generalizing a celebrated theorem of Magyar, Weyman and Zelevinsky. We give criteria that guarantee that a tree variety has a dense $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbit and provide many examples of tree varieties that do not have dense $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbits. We show that a triple of two-step flag varieties $$F(k_1, k_2; n)^3$$ has a dense $${{\,\mathrm{\mathbb {P}GL}\,}}(n)$$ orbit if and only if $$k_1 + k_2 \not = n$$ .