  We determine the global behavior of every C^2-solution to the two-dimensional
degenerate Monge-Ampere equation, u_{xx}u_{yy}-u_{xy}^2=0, over the finitely
punctured plane. With this, we classify every solution in the once or twice
punctured plane. Moreover, when we have more than two singularities, if the
solution u is not linear in a half-strip, we obtain that the singularities are
placed at the vertices of a convex polyhedron P and the graph of u is made by
pieces of cones outside of P which are suitably glued along the sides of the
polyhedron. Finally, if we look for analytic solutions, then there is at most
one singularity and the graph of $u$ is either a cylinder (no singularity) or a
cone (one singularity).
