Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between $0$ and $1$ or $1$ and $2$ they give an example where the line goes beween $2$ and $3$ and another one where the line goes between $3$ and $4$.

A variety $X$ over a field $K$ is of Hilbert type if $X\left(K\right)$ is not thin. We prove that if $f:X\to S$ is a dominant morphism of $K$-varieties and both $S$ and all fibers $f^{-1}\left(s\right)$, $s\in S\left(K\right)$, are of Hilbert type, then so is $X$. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.