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Let $X\rightarrow S$ be a holomorphic family of projective manifolds. My question is: can we find a Zariski open set $S'\subset S$ such that for any $t\in S'$, any given subvariety $Y\subset X_t$ with codimension $k$, there exists a relative subvariety $Z\subset X$ of codimension $k$ with $X_t\cap Z=Y $?

The motivation of this question is from Claire Voisin's proof of algebraic hyperbolicity of general hypersurfaces of degree large enough (in her article "On some problems of Kobayashi and Lang", proof of thm 3.3). The statement is as follows: Let $U$ be the Zariski open set parametrizing smooth hypersurfaces in $P^{n+1}$ of degree $d$. Let $\pi: X\rightarrow U, X\subset P^{n+1}\times U$ be the universal family. Assume for a general $X_t$ $(t\in U)$ there is given a subvariety $Y\subset X$ of codimension $k$. Then there exist a quasi-projective variety $B$, a dominant morphism $\phi:B\rightarrow U$, and a relative subvariety $Z\subset X_B$ of codimension $k$, where $X_B:=X\times_U B$, such that $Y\subset X_t$ is one fiber $Z_b$ for some $b\in B$.

My question is, how to prove the existence of this etale cover $\phi:B\rightarrow U$, and does this claim hold for any holomorphic family of projective manifolds?

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    $\begingroup$ No, that is not true, unless you allow $S'$ to be the empty set. Consider the family of polarized K3 surfaces of some genus $g$, and consider the family of $(-2)$-curves contained in fibers. $\endgroup$ Jun 2, 2015 at 14:32
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    $\begingroup$ @ Jason: Thanks. Can we add some conditions to make this statement to be true? In other word, when could this type of extension hold? $\endgroup$ Jun 2, 2015 at 14:43
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    $\begingroup$ "Can we add some conditions to make this statement to be true?" You could assume that $X$ is bimeromorphic to a trivial product $X_0\times S$. $\endgroup$ Jun 2, 2015 at 15:14
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    $\begingroup$ @RuadhaíDervan. That lemma only produces a Cartier divisor in $U$. It does not produce a Cartier divisor whose intersection with $X_t$ is a given Cartier divisor. Indeed, there can be no such theorem, as follows by considering $(-2)$-curves on K3 surfaces. $\endgroup$ Jun 2, 2015 at 17:10
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    $\begingroup$ @RuadhaíDervan: Actually, I might have spoken too soon. Certainly for a product $X_0\times S$, it is possible to extend subvarieties. However, already for the family of blowings up of $\mathbb{P}^2$ at $9$ points, I suspect that the rational curves of negative self-intersection exhibit the same type of behavior as with K3 surfaces. So I should not have suggested that there is a positive answer in the case of a "bimeromorphically trivial" fibration. $\endgroup$ Jun 2, 2015 at 18:15

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