In the article "On the Kodaira Dimension of the Moduli Space of Curves" by J. Harris and D. Mumford, to prove that $\overline{H}_{k,b}$ is proper over Spec $\mathbb{C}$, the authors refer to a "weak valuative criterion for properness." I am not familiar with and cannot find a reference to a "weak valuative criterion for properness." Why "weak..."?
What I am familiar with is the "valuative criterion for properness", used, for example, to prove that the Hilbert and Quot schemes are projective.
Does this "weak..." have any mathematical meaning?
