Irreducibility and dimension theorems for families of height 3 Gorenstein algebras
\(R\) is the ring \(k[x,y,z]\) where \(k\) is an algebraically closed field. The author shows that when a set \(D=(Q,P)\) of generator and relation degrees is given, the family \(\text{Gor}_D\) of Gorenstein algebras \(A=R/J\) with \(J\) having this set of generator and relation degrees is irreducible. By using deformation theory, one proves that when \(T\) is fixed the family \(\text{Gor}_T\) of Gorenstein algebras with Hilbert function \(T\) is irreducible. Then, one determines the maximum number of generators possible for an ideal determining a given Hilbert function. Finally, one investigates various methods for determining the dimension of \(\text{Gor}_T\), one makes a conjecture for this dimension and one proves this conjecture in many cases.
scientific article; zbMATH DE number 910264