  We study the wall-crossing diagrams for moduli spaces of coherent systems of
dimension one and order one on a smooth projective variety over the complex
numbers. We describe the right and left fibers in the wall-crossing diagrams.
We compute the topological Euler characteristic of these fibers in the
particular case when the variety is a quadric surface and the terms of the
Jordan-Holder filtration that are of order zero are supported on lines of
degree (0,1). We express the topological Euler characteristic of the moduli
spaces in terms of the topological Euler characteristic of certain Hilbert
schemes, in the particular case when the variety is a quadric surface and the
first Chern class of the coherent systems is of the form (2,r).
