Unobstructedness and dimension of families of Gorenstein algebras (Q2382026)

Let \(B=R/I_B\) be a graded Cohen-Macaulay quotient of codimension \(c\) of a polynomial algebra \(R\), \(n=\dim B\geq 2\) and \(K_B= \text{Ext}^c_R(B,R(-n-c))\) the canonical module of \(B\). Let \(M\) be a finitely generated graded \(B\)-module and \(\sigma : M(-s)\rightarrow B\) a morphism of graded \(B\)-modules such that \(\widetilde M\) (resp., \({\widetilde \sigma} : {\widetilde M}(-s)\rightarrow \widetilde B\)) is locally free of rank \(r\), \(1\leq r\leq n\), (resp., regular) in codimension \(<\max(r,2)\) on \(Y:=\operatorname{Proj}B\). \textit{J. O. Kleppe} and \textit{C. Peterson} [J. Algebra 238, No. 2, 776--800 (2001; Zbl 1028.13007)] proved that if \(({\bigwedge}^rM)^{\ast \ast} \simeq K_B(t)\) for some \(t\) and if \(({\bigwedge}^iM)^{\ast \ast}\) is Cohen-Macaulay for \(1\leq i\leq r/2\) (where ``\(\ast \)'' means ``\(\Hom_B(-,B)\)'') then \(A:=\operatorname{Coker}\sigma \) is Gorenstein. In the paper under review, which is a continuation of \textit{J. O. Kleppe} [Trans. Am. Math. Soc. 358, No. 7, 3133--3167 (2006; Zbl 1103.14005)] where one considers the case \(r=1\), the author uses deformation theory to vary \(B\), \(M\), \(\sigma \) in the above construction in order to see how large the stratum of the corresponding \(A\) in \(\text{GradAlg}^H(R)\) will be. Here \(\text{GradAlg}^H(R)\) denotes the scheme of graded quotients of \(R\) with Hilbert function \(H\). The dimension formulas he obtains are ``more computable'' for \(r=2\) or 3 and \(B\) licci (i.e., in the linkage class of a complete intersection). These formulas are particularly explicit when \(B\) is of codimension 2 and \(M\) is the normal module \(N_B=(I_B/I_B^2)^{\ast}\) or the first Koszul homology module \(\text{H}_1(I_B,R)\) associated to a minimal set of generators of \(I_B\). The main applications are for Gorenstein quotients of codimension 4 of \(R\), especially for arithmetically Gorenstein curves in \({\mathbb P}^5\).