**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!



Degenerations of smooth projective varieties
Q1: Sure, you can have embedded components. Project a curve in $P^3$ into a plane. Where the image crosses itself, you get embedded points.


You can improve connected to "set-theoretically equidimensional, and connected in codimension 1". 


I think the real moral should be that you shouldn't keep track just of the scheme structure when you're done degenerating a smooth variety, but of e.g. the induced log structure, and consider a moduli space of varieties + that extra structure. I should probably ask a question about that!


