Degenerate flag varieties: moment graphs and Schroder numbers

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schroder number and the Poincar, polynomial is given by a natural statistics counting the number of diagonal steps in a Schroder path. As an application we obtain a new combinatorial description of the large and small Schroder numbers and their q-analogues.