  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.
