  We discuss a paper of M. Green from a new algebraic perspective, and provide
applications of its results to level and Gorenstein algebras, concerning their
Hilbert functions and the weak Lefschetz property. In particular, we will
determine a new infinite class of symmetric $h$-vectors that cannot be
Gorenstein $h$-vectors, which was left open in a recent work of
Migliore-Nagel-Zanello. This includes the smallest example previously unknown,
$h=(1,10,9,10,1)$. As M. Green's results depend heavily on the characteristic
of the base field, so will ours. The appendix will contain a new argument,
kindly provided to us by M. Green, for Theorems 3 and 4 of his paper, since we
had found a gap in the original proof of those results during the preparation
of this manuscript.
