Coherent systems and BGN extensions on nodal reducible curves

Let $(C,\underset{‾}{w})$ be a polarized nodal reducible curve. In this paper we consider coherent systems of type $(r,d,k)$ on $C$ with $k<r$. We prove that the moduli spaces of $(\underset{‾}{w},\alpha )$-stable coherent systems stabilize for large $\alpha$ and we generalize several results known for the irreducible case when we chose a good polarization. Then, we study in details the components of moduli spaces containing coherent systems arising from locally free sheaves.