Question about intersection theory of algebraic varieties

Question

I have a basic question about the intersection theory of algebraic varieties. Let $X$ be a smooth projective variety defined over $\mathbb{C}$ of dimension greater than $2$. Let $D$ be an effective divisor on $X$ and $C$ be an irreducible curve such that $C\subset D$. Then can $D.C=0$? If not is there a counterexample?

Any comments regarding the question are very much appreciated.

Answer 1

Sure, take $X = (\mathbb P^1)^3, D = (\mathbb P^1)^2 \times \infty , C = \mathbb P^1 \times \infty \times \infty$. Then $D$ is an effective divisor, $C$ is a curve, $C \subset D$, $C$ is irreducible, but $C$ is rationally equivalent to $\mathbb P^1\times \infty \times 0$ which doesn't intersect $D$ and so $C\cdot D=0$.