Let $X$, $Y$ be abelian varieties and let $f:X\to Y$ be a morphism. They told me that we can define the pullback $f^*s$ of a global section $s\in\Gamma (L)$ where $L$ is an ample line bundle on $Y$, as $s$ composed with $f$. But how can I compose a global section with a morphism? It means that I can see a global section as a morphism? If yes, from where to where?
If it can be useful, I'm interested in particular in the case $Y=X/F$, $F\subset X$ finite group, $f=\pi$ étale morphism.
Thanks!
Recall that a section $s$ of $L$ is a map $s : Y \to L$ such that $\pi\circ s = \operatorname{id}_Y$. Given $f : X \to Y$, we can form the composition $s\circ f : X \to L$. Now $s\circ f$ is not a section of $L$ as $L$ is a bundle on $Y$, not $X$. However, we can form the pullback bundle $f^*L$, then $f^*s := s\circ f$ is a section of $f^*L$.