Hilbert's irreducibility theorem and jumps in the rank of the Mordell-Weil group

https://hu-berlin.zoom.us/j/61339297016

Let $k$ be a number field and $U$ a smooth integral $k$-variety.
Let $X \to U$ be an abelian scheme of relative dimension at least one.
We consider the set $U(k)_{+} \subset  U(k)$ of $k$-rational points  $m\in U(k)$ such that the 
Mordell-Weil rank of the fibre $X_{m}$  at $m$, which is an abelian variety over $k$, 
is strictly bigger than the Mordell-Weil rank of the generic fibre $X_{k(U)}$ over the
function field  $k(U)$.

 We prove:  if the $k$-variety $X$ is $k$-unirational, then  $U(k)_{+}$ is dense for the Zariski 
topology on $U$. If the $k$-variety $X$ is $k$-rational, then  $U(k)_{+}$ is not a thin set in $U$.
The second result leads us to a discussion of varieties over which Hilbert's irreducibility theorem holds.