Let $\pi:X^{e}\rightarrow Y^{d}(e\geq d)$ be a proper and dominant morphism of projective varieties over field $k$. Moreover, $Y$ is assumed to be smooth. Denote $\eta$ the generic point of $Y$, $X_\eta$ the generic fiber. For any point $x\in X_\eta$, the closure $\overline{\{x\}}$ in $X$ is a subvariety of dimension $d$, the restriction $\pi|_{\overline{\{x\}}}:\overline{\{x\}}\rightarrow Y$ is a finite morphism of degree $[\kappa(x):k(Y)]$. Now let $L$ be an invertible sheaf over $X$ and $\alpha\in A_{1}(Y)$ a cycle of curves on $Y$. I want to find the relationship between $c_1(\pi_*(L))\cdot \alpha$ and $(c_1(L)\cdot \pi^*(\alpha))|_{\overline{\{x\}}}$. If $d=e$, these two intersection numbers only have difference multiplying $[\kappa(x):k(Y)]$. But for the case $e>d$, I don’t know how to compare $\pi_*(L\otimes_{\mathscr{O}_X} \mathscr{O}_{\overline{\{x\}}})$ with $\pi_*(L)$, where $\mathscr{O}_{\overline{\{x\}}}$ is the structure sheaf of ${\overline{\{x\}}}$.
Thank you in advance if anyone could give any kind of advice.
