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:



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> 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.
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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.



Limits of reduced schemes question from Eisenbud and Harris
Such a subscheme is given by an ideal $I$ such that $m^4 \subset I \subset m^3$. In fact, any vector subspace in $m^3$ containing $m^4$ is an ideal (this is a simple exercise). Since $\dim O/m^3 = 15$ and $\dim O/m^4 = 35$, and we are interested in subspaces $I \subset O$ such that $\dim O/I = 21$, that is in subspaces $I/m^4$ of dimension $35 - 21 = 14$ of the space $m^3/m^4$ of dimension $35 - 15 = 20$. So, the family in question is just the Grassmannian $Gr(14,20)$. Its dimension is $14(20-14) = 14\cdot 6 = 84$.


