# Katz-Mazur chapter 1 AG questions

I was reading through Katz-Mazur "Arithmetic Moduli of Elliptic Curves", Chapter 1, and ran into some small issues (which might have a lot to do actually with notation). I think most of them are due to not being comfortable enough with the algebraic geometry notions involved. I managed to read on ignoring them but they are still bugging me. I apologize if the questions seems somewhat contentless.

Here is a link to the book. Below are my questions.

$$C/S$$ is a smooth curve over $$S$$ and $$D$$ is an effective Cartier divisor.

(1) Does $$\mathcal O_D$$ mean the structure sheaf of $$D$$ as a scheme, or is it $$\mathcal O_C/I(D)$$ as a sheaf on $$\mathcal O_C$$? I know they are roughly the same, but I am not comfortable enough in my algebraic geometry yet to treat them as the same (if I am wrong, please point out). In the latter case, (1') would the structure sheaf of $$D$$ be $$i^*(\mathcal O_D)$$, where $$i$$ is the inclusion $$D\hookrightarrow X$$, is that correct? (1'') Otherwise, if $$\mathcal O_D$$ meant the structure sheaf on $$D$$, would $$\mathscr L\otimes_{\mathcal O_C}\mathcal O_D$$ mean then $$\mathscr L\otimes_{\mathcal O_C}i_*(\mathcal O_D)$$?

(2) On page 9, we have

$$\mathscr L/\mathcal O\cong I^{-1}(D)/\mathcal O\cong I^{-1}(D)\otimes_{\mathcal O_C}\mathcal O_D\cong\mathscr L\otimes_{\mathcal O_C}\mathcal O_D=$$ an invertible $$\mathcal O_D$$-Module. Therefore $$D$$ is proper over $$S$$ if and only if the sheaf $$\mathscr L/\mathcal O$$ has its support proper over $$S$$.

Why is this so?

(3) Here $$X$$ is any scheme over $$S$$ and $$D,D'$$ effective Cartier divisors. On page 13,

Globally, in terms of representatives $$(\mathscr L,\ell)$$ for $$D$$ and $$(\mathscr L,\ell')$$ for $$D'$$, the condition $$D'\le D$$ is precisely that the global section $$\ell$$ of $$\mathscr L$$ vanish identically in $$\mathscr L|D'=\mathscr L\otimes_{\mathcal O_X}\mathcal O_{D'}$$.

Ok, so firstly here $$\mathscr L|D'=\mathscr L\otimes_{\mathcal O_X}\mathcal O_{D'}$$ really bugs me. From what I know, restriction of a sheaf to a subscheme is via pullback of the inclusion map and it gives a sheaf on the subscheme. So then the LHS would be a sheaf on $$D'$$ while the right hand side is a sheaf on $$X$$. There is an exact sequence below involving such terms and it is definitely meant as an exact sequence of sheaves on $$X$$, so... what exactly does $$\mathscr L|D'$$ denote here?

Secondly (3') What does it mean that $$\ell$$ vanishes in $$\mathscr L\otimes_{\mathcal O_X}\mathcal O_{D'}$$ and why is this implies by $$D'\le D$$? I would be grateful if someone could spell this out in some detail. I don't have any issue understanding the discussion of $$D'\le D$$ above in terms of their ideal sheaves but somehow I can't translate this into $$(\mathscr L,\ell)$$ language.

• $\mathscr O_D$ is the structure sheaf on $D$. It can be identified with a sheaf on $C$. Locally, what is going on is very simple: In the language of rings, if $C$ corresponds to $A$ and $D$ is given by the vanishing of $f$ so that $I(D) = (f) \subset \mathscr O_C = A$, then $\mathscr O_D$ corresponds to $A/(f)$ which is also $\mathscr O_C/I(D).$ Jun 19, 2018 at 11:46
• Does the notation $\mathcal{O}_C(-D)$ ring any bells? Maybe that will help you with relating the effective Cartier divisor with a line bundle plus an invertible section. Jun 19, 2018 at 12:55