Some algebraic consequences of Green's hyperplane restriction theorems

TL;DR

This paper explores algebraic implications of Green's hyperplane restriction theorems, applying them to level and Gorenstein algebras to identify new constraints on their Hilbert functions and Gorenstein h-vectors, including the smallest known example.

Contribution

It introduces a new algebraic perspective on Green's theorems and determines an infinite class of symmetric h-vectors that cannot be Gorenstein, addressing an open problem.

Findings

Identified a new infinite class of non-Gorenstein symmetric h-vectors

Determined the smallest example h=(1,10,9,10,1)

Extended Green's results with a new proof argument

Abstract

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.