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Let $S$ be a scheme, and $C$ be a smooth curve over $S$. It is known that any section $s\in C(S)$ of the curve defines a (relative) effective Cartier divisor of degree $1$, often denoted $[s]$ (see Katz and Mazur, page 8, page 10 in the pdf. See also this related question). I think I lack intuition about this construction in general, so I would to ask about the proper way to think about it.

Question 1: How can we describe this effective Cartier divisor $[s]$ in term of a closed subscheme of $C$? That is, what is the underlying topological space? Zariski locally on $S$, say if we consider $S=\operatorname{Spec}(R)$ and $C$ is covered by the open affines $U_i=\operatorname{Spec}(A_i)$, what does the equation defining $[s]$ look like? (Being an effective Cartier divisor, it is defined locally on $S$ by a single equation $f_i=0$, such that $A_i/f_iA_i$ is a flat $R$-module and $f_i$ is not a zero-divisor in $A_i$).

As far as the underlying topological space is concerned, it seems to me from the case where $S$ is the spectrum of a field, that it should be the image of $s:S\rightarrow C$. If this is true, what about the equation $f=0$ defining it?

Question 2: Now, let $p$ be a prime number and $e\geq 1$ any integer. In the particular case where $E$ is an elliptic curve over $S$ given with a section $0\in E(S)$, and if $S$ is an $\mathbb{F}_p$-scheme, how can one think of the effective Cartier divisor $p^e[0]$? I know it must be the kernel of the $e$-fold Frobenius morphism $F^e : E\rightarrow E^{(p^e)}$, but how can one see it/justify it easily? I fail to find any detailed justification of this statement.

I thank you very much for your help.

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    $\begingroup$ Any section of a morphism of schemes is a locally closed immersion (and closed if the morphism is separated), so the underlying topological space of of $[s]$ is just $s(S)$. To get a divisor, you also have to remember the scheme structure. [This has nothing to do with relative divisors, but rather with the equivalence between effective Cartier divisors and regularly immersed subschemes of codimension 1; try to write down some examples.] $\endgroup$ – user501746 Aug 7 '18 at 13:41

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