Ancestor ideals of vector spaces of forms, and level algebras

Let R = k[x1, . . . , xr] denote the polynomial ring in r variables over a field k, with maximal ideal M = (x1, . . . , xr), and let V ⊂ Rj denote a vector subspace of the space Rj of degree-j homogeneous elements of R. We study three related algebras determined by V . The first is the ancestor algebra Anc(V ) = R/V whose defining ancestor ideal V is the largest graded ideal of R such that V ∩ M j = (V ), the ideal generated by V . The second is the level algebra LA(V ) = R/L(V ) whose defining ideal L(V ), is the largest graded ideal of R such that the degree-j component L(V ) ∩ Rj is V ; and third is the algebra R/(V ). We have that L(V ) = V + M . 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 (Theorem 2.17). 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 (Theorem 2.24). We show that when r = 2 and k is algebraically closed the Hilbert function strata for each of the three algebras attached to V satisfy a frontier property that the closure of a stratum is the union of more special strata. In each case the family G(H) of all graded quotients of R having the given Hilbert function is a natural desingularization of this closure (Theorem 2.32). We then solve a refinement of the simultaneous Waring problem for sets of degree-j binary forms. Key tools throughout include properties of an invariant τ (V ), the number of generators of V ⊂ k[x1, x2], and previous results concerning the projective variety G(H) in [I2].