Proof of the Gorenstein Interval Conjecture in low socle degree

TL;DR

This paper proves the Gorenstein Interval Conjecture for low socle degrees (up to 5) in arbitrary codimension, revealing a structural property of Gorenstein Hilbert functions using algebraic and geometric methods.

Contribution

It establishes the Gorenstein Interval Conjecture for socle degree up to 5, providing new insights into the structure of Gorenstein Hilbert functions.

Findings

Proves GIC for socle degree e ≤ 5 in arbitrary codimension.

Uses constructive algebraic and geometric techniques.

Enhances understanding of Gorenstein Hilbert functions' structure.

Abstract

Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever $(1,h_1,\dots,h_i,\dots,h_{e-i},\dots,h_{e-1},1)$ and $(1,h_1,\dots,h_i+\alpha,\dots,h_{e-i}+\alpha,\dots,h_{e-1},1)$ are both Gorenstein Hilbert functions for some $\alpha \geq 2$, then $(1,h_1,\dots,h_i+\beta,\dots,h_{e-i}+\beta,\dots,h_{e-1},1)$ is also Gorenstein, for all $\beta =1,2,\dots,\alpha -1$. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case $e\le 5$, in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry.