Birational maps, PBW degenerate flags and poset polytopes

We extend the results on the graph closures of the birational maps between projective spaces and Grassmannians to the case of PBW degenerate flag varieties. The advantage of the PBW degenerate flags (as opposed to their classical analogues) is the existence of a large group of symmetries for the graph closures. We discuss the combinatorial, algebraic and geometric sides of the picture. In particular, we show that toric degenerations of Borovik, Sturmfels and Sverrisd\'ottir are still available in the general settings. We also derive a description of the graph closures for flag varieties in terms of quiver representations.