Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ such that their intersection is $k^n$ distinct points all lying inside $U$?
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2$\begingroup$ More or less, yes. You may not get $k^n$ points, since this number will depend on $\deg X$ with respect to $\mathcal{O}_X(1)$. For proving this, show that the set of points $(Z_1,\ldots,Z_n)\in |\mathcal{O}_X(k)|$ with non-distinct points as intersection is a proper closed subset and similarly, the ones with non-empty intersection with $X-U$ is also a proper closed subset. $\endgroup$– MohanCommented Apr 15, 2016 at 13:31
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