Coherent systems on curves of compact type

Let $C$ be a polarized nodal curve of compact type. In this paper we study coherent systems $(E,V)$ on $C$ given by a depth one sheaf $E$ having rank $r$ on each irreducible component of $C$ and a subspace $V \subset H^0(E)$ of dimension $k$. Moduli spaces of stable coherent systems have been introduced by King and Newstead and depend on a real parameter $\alpha$. We show that when $k \geq r$, these moduli spaces coincide for $\alpha$ big enough. Then we deal with the case $k=r+1$: when the degrees of the restrictions of $E$ are big enough we are able to describe an irreducible component of this moduli space by using the dual span construction.