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In this paper, we provide constructions to enumerate large numbers of CI-liaison classes. To this end, we introduce a liaison invariant and prove several results concerning it, notably that it commutes with hypersurface sections. This theory is applied to the CI-liaison classes of ruled joins of projective schemes, yielding strong obstructions for such joins to lie in the same liaison class. A second construction arises from the actions of automorphisms on liaison classes, allowing the enumeration of many liaison classes of perfect ideals of codimension at least three.

Since the 1970’s, great interest has been taken in the study of pure $O$-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.

We describe coefficient ideals for both $(x,y)$-primary monomial ideals in $k[x,y]$ and $\mathfrak{m}$-primary ideals in two-dimensio-nal regular local rings $(R,\mathfrak{m})$ by linking them to certain ideals of reduction number one. In the monomial case, we then explicitly determine the generators of a coefficient ideal by showing their symmetric relationship to the generators of the associated reduction number one ideal.