Hi,

I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, flat scheme of relative dimension $n$ and $Z\subset X_s$ be an equidimensional closed subscheme of the special fiber with dimension $d \lt n$. Given a point $t_{\eta}\in X(\eta)$ we know that this extends to a point $t\in X(S)$.

What can we say about subscheme $U\subset X_{\eta}$ of points specializing to $Z$?

Assume that $X\rightarrow S$ is very nice (like $X_{\eta}$ smooth and $X_s$ with semistable singularities) can I bound the dimension of $U$ in terms of the dimension of $Z$?