Reducible family of height three level algebras

Let R = k[x1, . . . , xr] be the polynomial ring in r variables over an infinite field k, and let M be the maximal ideal of R. Here a level algebra will be a graded Artinian quotient A of R having socle Soc(A) = 0 : M in a single degree j. The Hilbert function H(A) = (h0, h1, . . . , hj) gives the dimension hi = dimk Ai of each degree-i graded piece of A for 0 ≤ i ≤ j. The embedding dimension of A is h1, and the type of A is dimk Soc(A), here hj . The family LevAlg(H) of level algebra quotients of R having Hilbert function H forms an open subscheme of the family of graded algebras [Kle1, Kle4, I3] or, via Macaulay duality, of a Grassmannian [IK, ChGe, Kle4]. We show that for each of the Hilbert functions H1 = (1, 3, 4, 4) and H2 = (1, 3, 6, 8, 9, 3) the family LevAlg(H) has several irreducible components (Theorems 2.3A, 2.4). We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points (Theorem 2.3B). These were the first examples we obtained of multiple components for LevAlg(H) in embedding dimension three. We show that the second example is the first in an infinite sequence of examples of type three Hilbert functions H(c) in which also the number of components gets arbitrarily large (Theorem 2.10). The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors [BjI1] and also by J.-O. Kleppe [Kle4].