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Let $X$ be a quasiprojective variety over $\mathbf C$. Take the union of all projective subvarieties $W \subseteq X$ that have dimension at least $1$. Is the result Zariski closed?

(I was wondering this in the particular setting $X = \mathcal M_g$, where the projective subvarieties have been the subject of some study. But the general question seems natural as well.)

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No. For example take $X = \mathbb{A}^1 \times \mathbb{P}^1$ minus one point, say $(x,y)$. Then $W = (\mathbb{A}^1 \setminus x) \times \mathbb{P}^1$ is not closed.

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  • $\begingroup$ Oops, good point. $\endgroup$ – user47305 Jun 6 '19 at 23:07

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