  Let R be the polynomial ring in r variables over a field k, with maximal
ideal M and let V denote a vector subspace of the space of degree-j homogeneous
elements of R. We study three related algebras determined by V. The first is
the ``ancestor algebra'' whose defining ideal is the largest graded ideal whose
intersection with M^j is the ideal (V).
  The second is the ``level algebra'', whose defining ideal L(V) is the largest
graded ideal of R such that the degree-j component is V; and third is the
algebra R/(V).
  When r=2, we determine the possible Hilbert functions H for each of these
algebras, and as well the dimension of each Hilbert function stratum. We
characterize the graded Betti numbers of these algebras in terms of certain
partitions depending only on H, and give the codimension of each stratum in
terms of invariants of the partitions. When r=2 and k is algebraically closed
the Hilbert function strata for each of the three algebras satisfy a frontier
property that the closure of a stratum is the union of more special strata. The
family G(H) of all graded quotients of R having the given Hilbert function is a
natural desingularization of this closure.
