Boij-Soderberg theory for ideals generated by degree 2

TL;DR

This paper explores the limitations of Boij-Soderberg theory for degree 2 ideals, showing that certain degree sequences satisfying known conditions do not correspond to any ideal with a pure resolution.

Contribution

It constructs explicit degree sequences that meet necessary conditions but cannot be realized by any ideal with a pure resolution, revealing gaps in the theory.

Findings

Certain degree sequences are not realizable by ideals with pure resolutions

Necessary conditions are not sufficient for degree 2 ideals

Construction of counterexamples using generic initial ideals

Abstract

In Boij-Soderberg theory, it is known that for any degree sequence $\mathbf{d}$, there exists a finitely generated module that has a pure resolution of type $\mathbf{d}$. On the other hand, in the case of ideal, there are two necessary conditions for the degree sequence, which $\mathbf{d}$ satisfies them if there is an ideal that has a pure resolution of type $\mathbf{d}$. In this paper, by theory of generic initial ideals and Boij-Soderberg decompositions, we construct the degree sequence which satisfies these conditions but there is no such an ideal.