**Vague question.** Is there anything special about degenerations of smooth projective varieties (separating them from arbitrary projective schemes)?

**Precise setup.** Let $f:X\to Y$ be a projective flat morphism of algebraic schemes (say, over $\mathbb{C}$),
where $Y$ is an irreducible variety. Suppose that the generic fiber of $f$ is smooth and connected. Consider special fibers $X_y=f^{-1}(y)$, $y\in Y$. Of course, $X_y$ may be singular, reducible, and/or non-reduced. Zariski's main theorem implies $X_y$ is connected.
Can we say anything else?

**Specific Questions.** Can $X_y$ have embedded components? Can $X_y$ have non-constant regular functions? (Since $X_y$ is connected, such a regular function would
have to be nilpotent.)

**Reformulation.** Consider a Hilbert scheme of closed subschemes in ${\mathbb P}^N$
(with fixed Hilbert polynomial). There is an open subset in it corresponding to smooth
connected projective varieties. The question concerns the closure of this set.

*Remark.* I ran across this in a concrete situation, where the goal is to understand $Rf_*O_X$. But it seems that the question is quite natural, so perhaps the corresponding statements or counterexamples are well known... Any comments would be helpful!