Gorenstein in codimension 4: the general structure theory

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 $(k + 1) \times 2k$ matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety $\mathrm{SpH}_k \subset \mathrm{Mat}(k + 1, 2k)$. This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.