Betti strata of height two ideals
  We determine the codimension of the Betti strata of the family G(H) parametrizing graded Artinian quotients A of the polynomial ring R in two variables, having Hilbert function H. The Betti stratum G(B,H) parametrizes all such quotients having the graded Betti numbers determined by the relation degrees B. Our method is to identify G(B,H) as the product of determinantal varieties. We then use a result of M. Boij that reduces the calculation of codimension to showing that the most special stratum has the expected codimension. We also show that the closure of the Betti stratum is Cohen-Macaulay, and the union of lower strata.   As application, we determine the Hilbert functions possible for A, given the socle type; and we determine the Hilbert function of the intersection of t general enough level algebra quotients of R, each of a given Hilbert function. 
Anthony Iarrobino