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I always trusted the following quite vague statement:

If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say varying in disc, then the intersection number $D_1(t)\cdots D_k(t)$ can only increase under specialisation of $t$.

Without any flatness assumption on $D_i(t)$ and $X_t$ this is false, as shown in the post Semicontinuity of degree of fibers for a proper map

I guess, under an appropiate fltaness assumption, my belief follows from the standard semi-continuity theorem (Hartshorne III.11). However, I was not able to find any precise statement/reference.

I would like to have a precise statment about the semi-continuity of intersection numbers, with possibly a reference in the literature.

This should also fit with the question More upper/lower semi-continuous functions in (algebraic) geometry?

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    $\begingroup$ In $\mathbb{P}^1$ with homogeneous coordinates $[x,y]$, consider the closed subscheme in $\mathbb{P}^1\times \Delta$ with defining ideal $\langle x^2,tx\rangle$. That is a closed subscheme that is not flat over $\Delta$. However, its intersection with the fiber over every $a\in \Delta$ is a divisor in $\mathbb{P}^1$. Except if $a=0$, that divisor has degree $1$. For $a=0$, the divisor has degree $2$. Most algebraic geometers would not consider the closed subscheme to be a "family of divisors", i.e., they would impose a flatness hypothesis (or some hypothesis equivalent to flatness). $\endgroup$
    – Jason Starr
    Commented May 5, 2016 at 17:03
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    $\begingroup$ this example does not contradict a semi-continuity statement about intersection numbers. Anyway, I will make more precise my question $\endgroup$
    – Giulio
    Commented May 5, 2016 at 19:45
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    $\begingroup$ @JasonStarr Is it even simpler if we just take a single (closed) point in $P^1\times\Delta$? $\endgroup$
    – Fan Zheng
    Commented May 5, 2016 at 20:10
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    $\begingroup$ I am happy with the fact that the intersection number can go up, a bit less happy with the fact that it can go down. (I have in mind the example of the dimension of the fibre of a morphism: the dimension can only go up under specialisation, not down). For instance, take the divisor to be effective. I do not quite understand the example with divisor with negative self-intersection. $\endgroup$
    – Giulio
    Commented May 5, 2016 at 21:14
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    $\begingroup$ I just found another relevant post! I'll updated my question $\endgroup$
    – Giulio
    Commented May 6, 2016 at 12:51

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