Bezout thereom that says that two irreducible algebraic curves $C$ and $D$ in $\mathbb{P}^2_\mathbb{C}$ intersect at $nm$ points (counted with multiplicity), where $n$ and $m$ are the degrees of $C$ and $D$, respectively. So in this case we know that $C\cdot D=nm$ is finite.

The question is, if I change $\mathbb{P}^2_\mathbb{C}$ by any other non-singular projective algebraic surface $X$, may I find two irreducible algebraic curves $C,D \subset X$ such that $C \cdot D=\infty$? If yes, what can we say about $X$, $C$ and $D$?