Simple question about the definition of divisor Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{*}$ is the pull-back and $E$ a divisor on $C$, how can i define...--prophetes.ai

Simple question about the definition of divisor Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{}$ is the pull-back and $E$ a divisor on $C$, how can i define the divisor $\pi^{}(E)$?

It is essentially the preimage of $E$ but with multiplicities. If $p\in C$ is a point, then $$\pi^*(p):=\sum_{x\in\pi^{-1}(p)}(\mbox{mult}_x\pi)x$$ where $\mbox{mult}_x\pi$ is the local multiplicity of $\pi$ at $x$ (i.e., $\pi$ looks like $z\mapsto z^{\mbox{mult}_x\pi}$ at $x$). You can then extend this definition linearly to all divisors.

Simple question about the definition of divisor Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{*}$ is the pull-back and $E$ a divisor on $C$, how can i define the divisor $\pi^{*}(E)$?

Simple question about the definition of divisor Let $C$ a complex, compact riemann surface and $\pi:C^{'} \rightarrow C$ a generic cover of $C$. If $\pi^{}$ is the pull-back and $E$ a divisor on $C$, how can i define the divisor $\pi^{}(E)$?