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We work over a field $k$. Let $B$ be an algebraic variety over $k$. Suppose we are given a family of subvarieties of $\textbf P_k^n$ with base $B$, by which I mean a subvariety $V\subset B\times\textbf P_k^n$ together with the projection $\pi:V\to B$. The members of the family are the fibers $V_b\subset\textbf P_k^n$. Suppose we also have a subvariety $X\subset\textbf P_k^n$.

$\textbf{Question}\,\,1$: How to show that the locus $L=\{b\in B\,\,|\,\,V_b\cap X\neq \emptyset \}$ is closed in $B$?

I just observed that $\pi$ is a closed map (it is proper), and noticed that $L=\pi(Z)$ with $Z=\{(b,x)\in B\times X\,\,|\,\,x\in X\cap V_b\}$. But how to show then that $Z$ is closed in $B\times X$?

$\textbf{Question}\,\,2.$ Also, (forgetting about $X$) if we are given a second family $\pi':W\to C$, I would like to see that the locus $L'=\{(b,c)\,\,|\,\,V_b\cap W_c\neq\emptyset\}$ is closed in $B\times C$.

My problem here - and also above - is that I cannot translate in a nice way the nonempty condition, which is the only one I have. Moreover, if I try to make $L,L'$ explicit, I just come up with infinite unions of closed, while I would like intersections, for instance.

Thank you for any advise.

[Note: these problems come from Joe Harris, "Algebraic Geometry. A first course."]

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2 Answers

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For Question 1, observe that $Z=(B\times X)\cap V$. (Just notice that $x\in V_b$ means $(b, x)\in V$.)

Question 2: consider the map $$f : (B\times C)\times {\bf P}^n \to (B\times {\bf P}^n)\times (C\times {\bf P}^n), \quad (b,c,x)\mapsto ((b,x), (c,x)).$$ Then $L'$ is the image of $f^{-1}(V\times W)$ by the projection $(B\times C)\times {\bf P}^n\to B\times C$.

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Oh. So $x\in V_b\cap W_c$ means $((b,x),(c,x))\in V\times W$. Now I see. Thank you very much! –  Brenin Apr 25 '12 at 17:55
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Question 1
If $\rho:V\to \mathbb P^n_k$ is the other projection, you have $L=\pi(\rho^{-1} (X))$

Question 2
You have a morphism $p:B\times \mathbb P^n_k\times C\times \mathbb P^n_k\to \mathbb P^n_k\times \mathbb P^n_k$.
Consider the diagonal $\Delta \subset \mathbb P^n_k\times \mathbb P^n_k$ and its inverse image $D=p^{-1}(\Delta)\subset B\times \mathbb P^n_k\times C\times \mathbb P^n_k$.

Then if you look at $\pi\times \pi':V\times W\to B\times C$ you have
$$L'=(\pi\times \pi')(D)$$

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Thank you! it's very clear! –  Brenin Apr 25 '12 at 18:01
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