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In fact this is a difficult question in general, I believe. For example, if $X$ is a closed subvariety of ${\mathbb P}^n$, then asking for the minimal number of affine opens that cover the complement ${\mathbb P}^n$ P}^n\setminus X$is the same as asking for the minimal number of hypersurfaces whose (set-theoretic) intersection equals$X\$. I believe this question is open in general.

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In fact this is a difficult question in general, I believe. For example, if $X$ is a closed subvariety of ${\mathbb P}^n$, then asking for the minimal number of affine opens that cover the complement ${\mathbb P}^n$ is the same as asking for the minimal number of hypersurfaces whose (set-theoretic) intersection equals $X$. I believe this question is open in general.
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