Gorenstein in codimension 4 - the general structure theory

TL;DR

This paper extends the structure theory of Gorenstein ideals from codimension 3 to codimension 4, describing their resolutions via a matrix related to the Spin-Hom variety, advancing theoretical understanding.

Contribution

It provides a new structure theorem for codimension 4 Gorenstein ideals, generalizing Buchsbaum and Eisenbud's codimension 3 results.

Findings

Describes the projective resolution of codimension 4 Gorenstein ideals.

Establishes a correspondence with a matrix of first syzygies and the Spin-Hom variety.

Offers a theoretical framework for future applications.

Abstract

I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.