Multicomplex Configurations: a case study in Gorenstein Liaison

TL;DR

This paper introduces multicomplex configurations, a new class of projective varieties constructed from Artinian monomial ideals, demonstrating their algebraic properties and Gorenstein liaison relationships.

Contribution

It develops the theory of multicomplex configurations, linking geometric polarization and vertex decomposition to algebraic properties and liaison classes.

Findings

Ideals admit Gr"obner bases with specific initial ideals

Configurations are in the Gorenstein liaison class of a complete intersection

Conditions for preserving algebraic properties are established

Abstract

We introduce and investigate multicomplex configurations, a class of projective varieties constructed via specialization of the polarizations of Artinian monomial ideals. Building upon geometric polarization and geometric vertex decomposition, we establish conditions under which such configurations retain desirable algebraic properties. In particular, we show that, given suitable choices of linear forms for substitution, the resulting ideals admit Gr\"obner bases with prescribed initial ideals and are in the Gorenstein liaison class of a complete intersection.