The complex Grassmann manifolds G_{n,2} are of specific mathematical interest as the canonical actions of an algebraic torus C^n as well as the compact torus T^n on these manifolds are closely related to many important problems. One of them is moduli compactifications of compact complex algebraic curves of genus zero with n marked points, in particular those compactifications given by the weighted pointed stable genus zero curves.

Many structures on G_{n,2} can be assigned to these torus actions. There is the standard moment map u: G_{n,2} —> D_{n,2}$ for the hypersimplex D_{n,2} contained in R^n. A regular value, in the classical sense, of the map u is known to be a point x in D_{n,2} such that the stabilizer of any y in u^{-1}(x), for the T^n-action on G_{n, 2} is the diagonal circle S^1. The preimage u^{-1}(x) in G_{n,2} is a smooth submanifold and the orbit space u^{-1}(x)/T^n is a symplectic manifold, known as a symplectic reduction for the given T^n-action. In addition, the polytopes assigned to the strata on G_{n,2} defined by the C^n-action, give the chamber decomposition of D_{n,2}. It is proved that for any chamber C_{w} of maximal dimension n-1 all preimges u^{-1}(x) for x in C_{w} are diffeomorphic, leading to the smooth manifold F_{w} contained in G_{n,2}/T^n.

In this talk we discus recent results related to the problem of description of smooth manifolds u^{-1}(x) for the Grassmannians G_{n,2}. We study in detail the Grassmann manifold G_{4,2} and prove that u^{-1}(x) is diffeomophic to S^3\times T^2 for any regular value x in D_{4,2}.

In addition, since any of the smooth manifolds F_{w} is a moduli space of weighted pointed stable genus zero curves, we discuss such moduli spaces which can be obtained by the symplectic reduction on G_{n,2}. In this context, we prove that the associated Deligne-Mumford spaces and Losev-Manin spaces are symplectic reduction only for $n=4, 5$.