Let $X$ be a projective variety over $\mathbb{C}$. Let $X_1, X_2, \ldots$ be proper closed subsets. Then $\cup_i X_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement.

Assume that, for all $i$, we have that $\mathrm{codim}(X_i)\geq 2$.

Then, does there exist a smooth projective curve $C\subset X$ such that $C$ and $\cup_i X_i$ are disjoint?

The condition on the codimension is clearly necessary. (Take $X_1$ to be an ample divisor to see this.)

Let $X$ be a projective variety over $\mathbb{C}$. Let $X_1, X_2, \ldots$ be proper closed subsets of $X$. Then $\cup_i X_i \neq X(\mathbb{C})$. However, I am interested in a stronger statement.

Assume that, for all $i$, we have that $\mathrm{codim}(X_i)\geq 2$.

Then, does there exist a smooth projective curve $C\subset X$ such that $C$ and $\cup_i X_i$ are disjoint?

The condition on the codimension is clearly necessary (Take $X_1$ to be an ample divisor to see this).