Non-emptiness of moduli spaces of coherent systems

Let X be a general smooth projective algebraic curve of genus g>1. We prove that the moduli space G(\alpha:n,d,k) of $\alpha $-stable coherent systems of type (n,d,k) over X is empty if k>n and the Brill-Noether number is negative. Moreover, if the Brill-Noether number is positive and <g and for some $\alpha >0$, G(\alpha:n,d,k) is non-empty G(\alpha :n,d,k) is non-empty for all $\alpha >0$ and G(\alpha:n,d,k)= G(\alpha ':n,d,k) for all $\alpha ,\alpha '>0$ and the generic element is generated.