Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent sheaf on $X$, flat over $\mathrm{Spec}(A)$. What can we say about the Euler characteristic of $\mathcal{F} \otimes \mathcal{O}_{X_t}$, $\chi(\mathcal{F} \otimes \mathcal{O}_{X_t})$ as $t$ varies over points in $\mathrm{Spec}(A)$? Is it upper-semi continuous, lower-semi continuous or constant? We know that if $\pi$ is projective then the Euler characteristic remains constant.
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$\begingroup$ It is typically neither upper semicontinuous nor lower semicontinuous. $\endgroup$– Jason StarrCommented Feb 18, 2016 at 21:47
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$\begingroup$ @JasonStarr Is there any condition we can add on $\pi$ (not properness) which will ensure one of them? $\endgroup$– user45397Commented Feb 18, 2016 at 22:03
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$\begingroup$ @JasonStarr Is there any text or literature which deals with a similar question? $\endgroup$– user45397Commented Feb 18, 2016 at 22:25
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$\begingroup$ I am unaware of any reference that studies this without a properness hypothesis. $\endgroup$– Jason StarrCommented Feb 18, 2016 at 23:57
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I am not aware of such results in full generality, but I know that working without the properness assumtpion was in part the main motivation for Grothendieck to write SGA 2.
Let me focus on a related question (but not exactly the same). Let $\pi : X \rightarrow Y$ be a flat morphism of finite type with $Y$ a smooth variety over a field. Given a coherent sheaf $F$ on $X$ flay over $Y$, you would like to know if $h^i(X_t, F_t)$ might be upper semi-continous in some cases.
Essentially, what you have to answer are the following question:
Is the sheaf $R^i \pi_* F$ coherent?
If the answer to this question is yes, then I believe the semi-continuity of $h^i(X_t, F_t)$ is true under the flatness hypothesis (this becomes a linear algebra question if I remember correctly).
As Jason Starr mentionned, the well-known result is that the answer to this question is yes, if you assume $\pi$ proper.
If you want to drop the properness assumption, you have to add some other hypotheses for the coherence of $R^i \pi_* F$ to hold. (though I don't have a counter-example, I am pretty sure the cohrence does not hold without any assumption). In fact, many interesting results in SGA 2 address the coherence issue if you withdraw the properness hypothesis.
The price you have to pay is to add a depth hypothesis. In fact you will "compactify" $\pi$ from $\tilde{X}$ to $Y$. But you don't assume that $F$ comes from a coherent sheaf on $\tilde{X}$. What you want to know is when the sheaf $R^i j_* F$ is coherent (where $j : X \rightarrow \tilde{X}$ is the open immersion).
You have to make some assumptions on the depth of $F_x$ for $x \in X$.
Corollary $2.3$ and Theorem $3.1$ of expose $VIII$ in SGA 2 tell you what depth hypothesis you have to add to get some coherence results
