  Let $R={\sf k}[x,y,z]$, the polynomial ring over a field $\sf k$. Several of
the authors previously classified nets of ternary conics and their
specializations over an algebraically closed field. We here show that when $\sf
k$ is algebraically closed, and the Hilbert function sequence $T=(1,3^k,1),
k\ge 2$ (i.e. $T=(1,3,3,\ldots,3,1)$ where $k$ is the multiplicity of $3$) then
the family $G_T$ parametrizing graded Artinian algebra quotients $A=R/I$ of $R$
having Hilbert function $T$ is irreducible, and $G_T$ is the closure of the
family $\mathrm{Gor}(T)$ of Artinian Gorenstein algebras of Hilbert function
$T$. We then classify up to isomorphism the elements of these families
$\mathrm{Gor}(T)$ and of $G_T$. Finally, we give examples of codimension three
Gorenstein sequences, such as $(1,3,5,3,1)$, for which $G_T$ has several
irreducible components, one being the Zariski closure of $\mathrm{Gor}(T)$.
