Variation of moduli spaces of coherent systems of dimension one and order one

We study the wall-crossing for moduli spaces of coherent systems of dimension one and order one on a smooth projective variety over the complex numbers. We compute the topological Euler characteristic of the moduli spaces in the particular case when the variety is a quadric surface, the first Chern class of the coherent systems is of the form (2,r) and the second Chern class is bounded from below by 3r + 1 and also by 4r - 8.