  Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an
infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \emph{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 $H(A)=(h_0,h_1,...
,h_j)$ gives the dimension $h_i=\dim_k A_i$ of each degree-$i$ graded piece of
$A$ for $0\le i\le j$. The embedding dimension of $A$ is $h_1$, and the
\emph{type} of $A$ is $\dim_k \Soc (A)$, here $h_j$. The family $\Levalg (H)$
of level algebra quotients of $R$ having Hilbert function $H$ forms an open
subscheme of the family of graded algebras or, via Macaulay duality, of a
Grassmannian.
  We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and
$H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian
algebras of Hilbert function $H$ has several irreducible components. 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. These were the first examples we obtained of multiple components for
$\Levalg(H)$ in embedding dimension three.
  We also 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 of LevAlg(H) gets arbitrarily large.
  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 and also by J.-O. Kleppe.
