TITLE: Degenerations of smooth projective varieties
QUESTION [9 upvotes]: 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!

REPLY [4 votes]: A partial answer: if $X$ is normal and $Y$ is smooth of dimension $1$, then $X_y$ only have constant regular functions ($X\to Y$ is cohomologically flat in relative dimension $0$). This is proved in Raynaud: Spécialisation du foncteur de Picard, Prop. 6.4.2 (use the characteristic 0 hypothesis here, otherwise it is false even when $X$ is also smooth.) It is also proved in the begining of ''Surfaces fibrées en courbes de genre deux'', Lecture Notes in Math. 1137 (1985) by Gang Xiao. 
Add: and of course in this situation $X_y$ has no embedded point as $X$ is (S$_2$).