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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?

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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.)

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  • $\begingroup$ Thank you for the nice answer. Is then the only hope to be dealing with an adjoint linear series, and trying to use one of the versions of invariance of plurigenera? $\endgroup$
    – Stefano
    Nov 23, 2016 at 16:24
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    $\begingroup$ @Stefano I can't think of many other cases where I can see a way to make it work. I guess another hope is that your bundle is actually ample in the special fibers! $\endgroup$
    – user47305
    Nov 29, 2016 at 16:30

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