Green's theorem and Gorenstein sequences

TL;DR

This paper explores the implications of extremal cases in Green's Hyperplane Restriction Theorem for graded algebras, extending the theorem and linking it to Macaulay's theorem to classify certain Gorenstein sequences.

Contribution

It extends Green's theorem from plane curves to hypersurfaces and connects it to Macaulay's theorem under a Lefschetz condition, leading to classification of specific Gorenstein sequences.

Findings

Proved that (1,19,17,19,1) is not a Gorenstein sequence.

Extended Green's theorem to hypersurfaces in linear spaces.

Classified Gorenstein sequences of the form (1,a,a-2,a,1).

Abstract

We study consequences, for a standard graded algebra, of extremal behavior in Green's Hyperplane Restriction Theorem. First, we extend his Theorem 4 from the case of a plane curve to the case of a hypersurface in a linear space. Second, assuming a certain Lefschetz condition, we give a connection to extremal behavior in Macaulay's theorem. We apply these results to show that $(1,19,17,19,1)$ is not a Gorenstein sequence, and as a result we classify the sequences of the form $(1,a,a-2,a,1)$ that are Gorenstein sequences.