Let $H_{P,n}$ be the Hilbert scheme of subschemes of $\mathbb{P}^n(\mathbb{C})$ with Hilbert polynomial $P\in\mathbb{Q}[t]$, and let $U_{P,n}\to H_{P,n}$ be the flat universal family. Are there $n,P$ and closed points $y_0,y_1$ in the same component of $H_{P,n}$ such that the fiber $U_{P,n,y_0}$ is equidimensional (meaning all components have the same dimension) and reduced, and the fiber $U_{P,n,y_1}$ is not equidimensional?
Comments: in simple examples (see e.g. Harris, Algebraic geometry, a first course, chapter 21, in particular the section on curves of degree 2), when a component merges with a higher dimensional component under a flat deformation, it still leaves behind a trace in the form of nilpotents in the structure sheaf of the limit. For example, one can deform the union of a line $l\subset\mathbb{P}^2(\mathbb{C})$ and a point $p\not\in l$ (with the reduced scheme structure) to a scheme $X$ such that the underlying topological space is the same as for $l$, but there are nilpotents in some $\mathcal{O}_{X,p_0},p_0\in X$. The question is whether or not this happens in general.
