TITLE: Projective subvarieties of a quasiprojective variety
QUESTION [5 upvotes]: 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.)

REPLY [10 votes]: 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.