Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two

Abstract

We let A=R/I be a standard graded Artinian algebra quotient of R=k[x,y], the polynomial ring in two variables over a field k by an ideal I, and let n be its vector space dimension. The Jordan type Pℓ of a linear form ℓ∈A1 is the partition of n determining the Jordan block decomposition of the multiplication on A by ℓ – which is nilpotent. The first three authors previously determined which partitions of n=dimk⁡A may occur as the Jordan type for some linear form ℓ on a graded complete intersection Artinian quotient A=R/(f,g) of R, and they counted the number of such partitions for each complete intersection Hilbert function T[1].We here consider the family GT of graded Artinian quotients A=R/I of R=k[x,y], having arbitrary Hilbert function H(A)=T. The Jordan cell V(EP) corresponding to a partition P having diagonal lengths T is comprised of all ideals I in R whose initial ideal is the monomial ideal EP determined by P. These cells give a decomposition of the variety GT into affine spaces. We determine the generic number κ(P) of generators for the ideals in each cell V(EP), generalizing a result of [1]. In particular, we determine those partitions for which κ(P)=κ(T), the generic number of generators for an ideal defining an algebra A in GT. We also count the number of partitions P of diagonal lengths T having a given κ(P). A main tool is a combinatorial and geometric result allowing us to split T and any partition P of diagonal lengths T into simpler Ti and partitions Pi, such that V(EP) is the product of the cells V(EPi), and Ti is single-block: GTi is a Grassmannian.