My question pertains to exercise II-16 in Eisenbud and Harris' "The geometry of Schemes". For an algebraically closed field $K$ the question is as follows:

Consider zero-dimensional subschemes $\Gamma \subset \mathbb{A}_K^4$ of degree 21 such that $$V(m^3)\subset\Gamma \subset V(m^4)$$ where $m$ is the maximal ideal of the origin in $\mathbb{A}_K^4$. Show that there is an 84-dimensional family of such subschemes, and conclude that in general one is not a limit of a reduced scheme.

What does it mean for a family of subschemes to have dimension 84? I can only think it means up to isomorphism there are 84 such subschemes, but this doesn't seem to work.

`$p_1: X \times \mathbb{A}^4 \to X$`

then $\pi^{-1}(x)$ for $x \in X$ (a closed point), which is naturally a closed subscheme of$\mathbb{A}^4$, should be a subscheme of the form $\Gamma$. Moreover, all these fibres should be distinct. The technical way to say this is that the corresponding Hilbert scheme has dimension at least $84$. – ulrich Jul 12 '11 at 18:06