  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 Schr\"oder number and the Poincar\'e polynomial is
given by a natural statistics counting the number of diagonal steps in a
Schr\"oder path. As an application we obtain a new combinatorial description of
the large and small Schr\"oder numbers and their q-analogues.
