On \alpha-stability and linear stability of generated coherent systems

There is a well studied notion of GIT-stability for coherent systems over curves, which depends on a real parameter $\alpha$. For generated coherent systems, there is a further notion of stability derived from Mumford's definition of linear stability for varieties in projective space. Let $\alpha_S$ be close to zero and $\alpha_L \gg 0$. We show that a generated coherent system which is $\alpha_S$-stable and linearly stable is $\alpha_L$-stable, and give examples showing that without further assumptions, there are no other implications between these three types of stability. We observe that several of the systems constructed have stable dual span bundle, including systems which are not $\alpha$-semistable for any value of $\alpha$. We use this to prove a case of Butler's conjecture for systems of type $(2, d, 5)$.