Betti strata of height two ideals

AbstractLet R=k[x,y] denote the polynomial ring in two variables over an infinite field k. We study the Betti strata of the family G(H) parametrizing graded Artinian quotients of R=k[x,y] having given Hilbert function H. The Betti stratum Gβ(H) parametrizes all quotients A of having the graded Betti numbers determined by H and the minimal relation degrees β, with βi=dimkTor1R(I,k)i. We recover that the Betti strata are irreducible, and we calculate their codimension in the family G(H).TheoremThe codimension of Gβ(H) in G(H) satisfies, letting νi=#{generatorsofdegree i}, and βi=#{relationsofdegree i}, codGβ(H)=∑i>μ(βi)(νi).Here μ=min{i|Hi