I am studying the definition of Chow variety on Kollar's Rational Curves on Algebraic Varieties, and I am having some trouble in understanding Definition 3.9.
Here we have a proper morphism of schemes $g_i:U_i\rightarrow W$ where $U$ is irreducible and $W$ is reduced. There is an open subset $W_i\subset g_i(U_i)\subset W$ such that $g_i$ is flat over it and of relative dimension $d$. Now he takes $T$ to be the spectrum of a DVR and a morphism $h:T\rightarrow W$ which sends the closed point $T_0\mapsto w\in W$ and the generic point to a point in $W_i$. Now let $h^\ast U_i$ be the pullback of $U_i$ via $h$ and define $J\subset\mathcal{O}_{h^\ast U_i}$ to be the ideal given by functions whose support is contained in the special fiber of $h^\ast U_i$ over $T$.
Set $f:(U_i)'_T=\text{Spec}(\mathcal{O}_{h^\ast U_i}/J)\rightarrow T$: it is a flat map by construction so the central fibre $Z_0$ has pure dimension $d$. Then one names the cycle theoretic fibre of $g_i$ at $w$ along $h$ to be the cycle $[Z_0]\in Z_d(g_i^{-1}(w)\times_w T_0)$.
What I think I am not understanding properly is how to picture the scheme $(U_i)'_T$. Moreover, and I think that this is the important question, I don't understand what is the point behind the construction of $(U_i)'_T$. I mean, why can't we just take the special fibre of $h^\ast U_i\rightarrow T$ and define its fundamental cycle to be the cycle theoretic fibre of $g_i$ at $w$ along $h$?
(Five years past the point of this being helpful, but I found this while trying to answer the same question myself and thought I'd leave the answer I eventually came up with:)
The intuition for $(U_i)_T'$, and the difference between your suggested construction and the construction given in Kollár, is in how it treats components of $h^* U_i$ lying over the special fiber of $T$. This can be seen in a simple example, where we take $T=W=\mathrm{Spec}\ k[[t]]$ and $U=\mathrm{Spec}\ k[[t][x]/(tx)$ (this isn't really proper, but you can compactify it to be proper and everything else works the same). Since $T=W$, $h^* U$ is just $U$. Note that $U\to W$ is not flat, since the generic fiber is $0$-dimensional and the special fiber is an $\mathbb A^1$. The ideal $J$ here is $(x)$ (since where $t\neq 0$, $x=xtt^{-1}=0$), and thus $U_T'=\mathrm{Spec} k[[t]]$. That is, $U'_T$ is the closed subset of $U$ obtained by discarding the vertical fiber lying over the closed point of $T$, so now $U'_T\to W$ is flat (in fact, an isomorphism).
So, in this case, the resulting special fiber of $U'_T$ over the closed point of $T$ is just a point, while the special fiber of $U\to T$ over the closed point is a curve. Away from this fiber, both families are $0$-dimensional. The cycle-theoretic fiber "knows" that it should be a limit of $0$-dimensional cycles (thus $0$-dimensional), while the scheme-theoretic fiber can jump dimensions arbitrarily.