Let $\pi: X \rightarrow Y$ be a morphism of varieties. Suppose $s: Y \rightarrow X$ is a section of $\pi$, that is, $\pi \circ s = id$. I came across with the following identity: $$s^{\ast}(s(Y)) = \pi_{\ast}[s(Y) \cdot s(Y)].$$ I tried to get it using projection formula, but I did not have success. Maybe (or in fact), it is because I am missing something in the article that I am reading, for example $\pi$ is a $\mathbb{P}^{1}$-bundle. If this is the case, does the identity above hold? I think so, but I am not able to prove it. In fact, I think a more general identity holds when $\pi$ is a $\mathbb{P^{n}}$-bundle. May anyone help or give me references?
Thank you.