Polarization and Gorenstein liaison
Sara Faridi, Patricia Klein, Jenna Rajchgot, Alexandra Seceleanu
TL;DR
The paper investigates the central glicci question in Gorenstein liaison by linking monomial and Gröbner-theoretic perspectives through polarization. It develops a lifting framework showing that basic double G-links on polarizations descend to the original monomial ideals, and proves that polarizations of several natural monomial classes are glicci via vertex-decomposable Stanley–Reisner complexes. It then extends these ideas to geometric polarization for Gröbner bases, establishing a polarLink bridge between elementary G-biliaisons on polarized ideals and those on the originals. The work unifies vertex decomposition, polarization, and geometric vertex decomposition to transfer glicci properties across polarizations and their Gröbner bases, broadening the scope beyond squarefree monomial ideals. Together, these results advance strategies for proving glicci status and deepen connections between combinatorial and geometric liaison theories.
Abstract
A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.