Some algebraic consequences of Green’s hyperplane restriction theorems

We discuss Green’s paper [11], 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 the recent work [19], . 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 contains a new argument, kindly provided to us by M. Green, for Theorems 3 and 4 of [11], since we had found a gap in the original proof of those results during the preparation of this manuscript.