Abstract
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
α
-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
α>0
, G(\alpha:n,d,k) is non-empty G(\alpha :n,d,k) is non-empty for all
α>0
and G(\alpha:n,d,k)= G(\alpha ':n,d,k) for all
α,
α
′
>0
and the generic element is generated.