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 $Z$?



scheme of generalizations
Suppose for simplicity that $\mathcal O$ is a henselian (e.g., complete) DVR (otherwise, a point of $X\_\eta$ may have several specializations in $X\_s$) with field of fractions $K$. Consider $X=\mathbb P^1\_S$ parametriezd by a rational function $t$, and let $Z$ be the single point $t=0$ in the closed fiber. The the set $U$ is the open disc $|t|<1$ in the generic fiber $\mathbb P^1\_K$ (in more algebraic terms, they correspond to maximal ideals $P(t)K[t]$ with $P(t)\in \mathcal O[t]$ monic and $P(0)\in \mathfrak m$ the maximal ideal of $\mathcal O$). So it is not a scheme, but a rigid analytic subspace.


