Let $X$ be a projective, smooth curve over an algebraically closed field, $Y = X\times X$ and $\mathcal{l}= pt\times X$ where $pt$ is a closed point of $X$. Can one say that the intersection number $\mathcal{l}^2$ is equal to zero by arguing that applying a translation to $\mathcal{l}$ moves it in its equivalence class so that they do not meet?
Is there a better way to see this? I am using the "usual" definition of the intersection number as found in Hartshorne Ch. V.
Thanks in advance!