Polarization and Gorenstein liaison

The 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 indeed it can be after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$. 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.  After reviewing the key aspects of liaison and polarization, we will describe conditions that allow for the lifting of a basic double G-link induced from a vertex decomposition of the Stanley--Reisner complex of the polarization of an ideal $I$ to a basic double G-link of $I$ itself.  We will introduce a notion of polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a result for geometric vertex decomposition and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage. Time permitting, we will give an application to vertex decompositions of Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals and of artinian monomial ideals, extending a recent result of Fl{\o}ystad and Mafi.

The work described in this talk is joint with Sara Faridi, Jenna Rajchgot and Alexandra Seceleanu. 

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