Rings of invariants and linkage of determinantal ideals
In this paper a criterion is given for when a ring of invariants under a finite Abelian group action is regular. The criterion involves the depth of certain conormal modules and the first Koszul homology, and is mainly interesting as a negative condition since it yields strong obstructions for deforming and linking rings of invariants which are not regular. As a first application the non-Cohen-Macaulay loci for conormal modules of symmetric determinantal ideals are computed. These results are used to show some ''rigid'' behavior of (symmetric) determinantal ideals under linkage (excluding the trivial cases where the ideals are complete intersections, or have grade 2). For example, if two determinantal ideals I and J, one of which is generic, belong to the same linkage class, then I and J have to be isomorphic; or let I be a symmetric determinantal ideal, let J be a determinantal ideal, and assume that I or J is generic, then I and J are in different linkage classes. A further application is the construction of infinitely many arithmetically Gorenstein curves in \({\mathbb{R}}^ 5\) (arithmetically Cohen-Macaulay curves in \({\mathbb{R}}^ 4)\) which are projectively equivalent but belong to different linkage classes, contrasting the well known fact that there exists only one linkage class of arithmetically Gorenstein curves in \({\mathbb{R}}^ 4\) (arithmetically Cohen-Macaulay curves in \({\mathbb{R}}^ 3)\).
conormal modules of symmetric determinantal ideals
scientific article; zbMATH DE number 3904754