Fix an algebraically closed field $k$. Let $X$ and $Y$ be proper varieties over $k$. If there is a connected scheme $B$ of finite type over $k$ such that $X$ and $Y$ embed in a proper flat family over $B$ is there also an irreducible such scheme?



Deformation equivalent varieties over an irreducible base
Not in general. Perhaps the simplest example is given by the Hilbert scheme of curves $C$ of degree 3 in $\mathbb{P}^3$ with $\chi (\mathscr{O}\_C)=1$. This has 2 components, one (of dimension 12) corresponding to twisted cubics and the other (of dimension 15) parametrizing the union of a plane cubic and a point in $\mathbb{P}^3$. They meet transversally along a 11-dimensional variety corresponding to plane cubics with an embedded point. It is easy to show that you cannot specialize a plane cubic + a point outside to a twisted cubic, yet they are deformation equivalent.


Reference: R. Piene, M. Schlessinger: *On the Hilbert Scheme Compactification of the Space of Twisted Cubics*
Amer. J. of Math.
**107**, no. 4 (1985), pp. 761-774.


