  We give necessary and sufficient conditions for moduli spaces of semistable
chains on a curve to be irreducible and non-empty. This gives information on
the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the
irreducible components of moduli of systems of Hodge bundles on curves. As we
do not impose coprimality restrictions, we can apply this to prove
connectedness for moduli spaces of U(p,q)-Higgs bundles.
