# Homogeneous Liaison and the Sequentially Bounded Licci Property

## TL;DR

This paper constructs examples of homogeneous licci ideals that are not sequentially bounded licci, disproving a conjecture that all such ideals satisfy the sequentially bounded condition.

## Contribution

It provides the first known examples of homogeneously licci ideals that are not sequentially bounded licci, answering a key open question.

## Key findings

- Not all homogeneous licci ideals are sequentially bounded licci.
- The structure of Betti tables is crucial in distinguishing these classes.
- The results impact the understanding of linkage classes in polynomial rings.

## Abstract

In CI-Liaison, significant effort has been made to study ideals that are in the linkage class of a complete intersection, which are called licci ideals. In a polynomial ring, recently E. Chong defined a "sequentially bounded" condition on the degrees of the forms generating the regular sequences, and used this condition to find a large class of licci ideals satisfying the Eisenbud-Green-Harris Conjecture (among them, grade $3$ homogeneous Gorenstein ideals). He raised the question of whether all homogeneous licci ideals are sequentially bounded licci. In this paper we construct a class of examples that are homogeneously licci, but not sequentially bounded licci, thus answering his question in the negative. The structure of certain Betti tables plays a central role in our proof.