Given a family of divisors $D\_t$ on varieties $X\_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D\_0$ is big, while for general $t$ $D\_t$ has negative Kodaira dimension).


Can we argue any openness if we know a bit more? Say that the base of the family is $U$, and for a countable set $\lbrace t\_i \rbrace$ such that $\overline{\lbrace t\_i \rbrace}=U$ we know that $D\_{t\_i}$ is big. Does then the general $D\_t$ have to be big?



Big divisors in family
Sadly, this isn't the case. The problem, basically, is that the effective cone can grow larger countably many times in a family: there is a countable set of line bundles which generically don't have sections, but each one does over some closed subset of the base. It's possible that when the effective cone jumps like this, there is some class that ends up being in the interior (hence big) every single time.


One example is worked out by Pan and Shen here: <https://arxiv.org/abs/1309.7535> . It's the same example where nefness fails countably many times in a family: for that class, it's very generally not big, but every time it fails to be nef (because there is an extra effective -2-curve class on which it's negative), it *is* big, since the effective cone gets bigger and it lands in the interior.


(I'm not sure whether examples are known for $\mathbb Q$-divisor classes, but one probably expects them to exist.)


