Artinian Gorenstein algebras of embedding dimension four and socle degree three

Abstract

We prove that in the polynomial ring ${Q=\kk[x,y,z,w]}$, with $\kk$ an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq (x,y,z,w)^2$ can be obtained by \emph{doubling} from a grade three perfect ideal $J\subset I$ such that $Q/J$ is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the \mbox{$Q$-module} $Q/I$ can be completely described in terms of a graded minimal free resolution of the \mbox{$Q$-module} $Q/J$ and a homogeneous embedding of a shift of the canonical module $\omega_{Q/J}$ into $Q/J$.