Fix an integer $n \ge 5$. Let $\mathcal{V}$ be a *countable* collection of closed subvarieties of $\mathbb{P}^n\_{\mathbb{C}}$ of codimension at least $2$. Choose a point $p \in \mathbb{P}^n$. Does there exist a curve $C$ (affine or projective) containing the point $p$ and not intersecting any subvariety $V \in \mathcal{V}$ away from $p$ i.e., for any $V \in \mathcal{V}$, $C \cap V$ is either $p$ or $\emptyset$?


**EDIT** The underlying field is $\mathbb{C}$.



Moving lemma for countable collection of subvarieties
Consider the case when $C$ is a line through $p$. Lines through $p$ correspond to points of $\mathbb P^{n-1}$, and this gives a projection map $\mathbb P^n \setminus p \to \mathbb P^{n-1}$ Then $C$ intersects $V$ if and only if the image of $V \setminus p$ under the projection map doesn't contain the point corresponding to $C$. Since $V$ has codimension at least $2$, the image under the projection map has codimension at least $1$.


The countable union of codimension $1$ subvarieties cannot contain all the points for the usual reasons (you can prove this with measure theory, Baire category theory, or purely algebraically by induction). So there must exist such a line $C$.


