# Smooth special fiber implies smooth generic fiber when proper and flat

Let $$f : X \rightarrow S$$ be a morphism between schemes, with $$f$$ locally of finite presentation, proper and flat, and $$S$$ the spectrum of a discrete valuation ring with closed point $$s$$.

Assume that the special fiber $$X_s$$ is smooth. Is the generic fiber smooth?

In other words: is it impossible to have a flat (algebraic) family of proper singular varieties degenerating into a smooth variety?

It is sufficient (and necessary because $$S$$ has just two points: the special point $$s$$ and the generic point) to show that $$f$$ is smooth. By this lemma (or EGA IV, 4, 17.5.1), we use local finite presentation and smoothness of the special fiber to conclude that $$f$$ is smooth on an open $$U \subset X$$ containing the special fiber $$f^{-1}(s)$$.
Since $$f$$ is proper, we conclude that $$U = X$$, so in particular the generic fiber is smooth. Indeed, $$X \setminus U$$ is closed; if it were non empty, then the properness of $$f$$ implies that $$f(X \setminus U) \subset S$$ is closed and non-empty, and since $$S$$ is a discrete valuation ring, $$f(X \setminus U)$$ has to be equal to $$s$$ or $$S$$. In both cases $$s \in f(X \setminus U)$$, which implies that $$X \setminus U$$ meets $$f^{-1}(s)$$, hence a contradiction.