Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a *generic* point on $B$. Denote by $\mathcal{X}\_K$, the pull-back of $\mathcal{X}$. This is flat, projective on $K$. It will then follow that $\mathcal{X}\_{K\_{\mathrm{red}}}$, the associated reduced scheme, is also flat, projective scheme over $K$. Assume that there exists a subfamily of $\pi$, say $\pi':\mathcal{X}' \to B$, a flat family of projective schemes satisfying $\mathcal{X}'\_b \subset \mathcal{X}\_b$ for all $b \in B$ and $\mathcal{X}'\_K = \mathcal{X}\_{K\_{\mathrm{red}}}$. Does this mean that for a *general* closed point $b \in B$, the fiber $\mathcal{X}'\_b=\mathcal{X}\_{b\_{\mathrm{red}}}$, the associated reduced scheme? If not true in general, is there any known condition on $B$ or $\pi$ under which this could hold true?



Generic vs General property of reducedness in a family of projective schemes
This is true if you assume moreover that $(\mathcal{X}\_K)\_{red}$ is *geometrically* reduced -- in particular, in characteristic $0$. First of all, note that set-theoretically $\mathcal{X}'\_b=\mathcal{X}\_b$ for all $b$ in $B$ : since $\pi $ open, $\pi (\mathcal{X}-\mathcal{X}')$ is an open subset of $B$ which does not contain the generic point, therefore it is empty. Now the subset $U\subset B$ of points $b$ such that $\mathcal{X}'\_b$ is geometrically reduced is open in $B$ (EGA IV, Thm. 12.2.4), so for $b\in U\ $ $\mathcal{X}'\_b$ is reduced, and therefore equal to $(\mathcal{X}\_b)\_{red}$.


I am not an expert in characteristic $p$ but I suspect there might be counter-examples if one does not assume that $(\mathcal{X}\_K)\_{red}$ is geometrically reduced.


