TITLE: Really rigid varieties
QUESTION [7 upvotes]: Are there (complex, projective) varieties $X_0$ with the following property?

Every (flat, say) family $\mathfrak X\to T$ over a reasonable and reasonably big class of bases with a fiber over a closed point equal to $X_0$ is a trivial family, in the sense that $\mathfrak X$ is $T\times X_0$.

If one asks this with $T$ restricted to local Artinian rings, say, one gets the class of (infinitesimally) rigid varieties, but I wonder if there are "globally rigid" examples.
Later: As Francesco and unknowngoogle observe, it is quite unreasonable to ask for such strong rigidity... :)

REPLY [12 votes]: If $X_0$ is projectively embedded, we can use a Gr\"obner basis for its defining ideal to make a Rees family over ${\mathbb A}^1$ most of whose fibers are $X_0$, but whose central fiber is a monomial scheme. So $X_0$ must already be a monomial scheme. Since you said it's a variety, it must be a projective space. If you then allow me to reembed (by a Veronese), I could break that projective space, too, unless it's a point. So my answer: $X_0$ must be a point.